![Prime Number Theory and the
Riemann Zeta-Function
D.R. Heath-Brown
1 Primes
An integer p ∈ N is said to be “prime” if p = 1 and there is no integer n
dividing p with 1 n p. (This is not the algebraist’s definition, but in our
situation the two definitions are equivalent.)
The primes are multiplicative building blocks for N, as the following crucial
result describes.
Theorem 1. (The Fundamental Theorem of Arithmetic.) Every n ∈ N
can be written in exactly one way in the form
n = pe1
1 pe2
2 . . . pek
k ,
with k ≥ 0, e1, . . . , ek ≥ 1 and primes p1 p2 . . . pk.
For a proof, see Hardy and Wright [5, Theorem 2], for example. The
situation for N contrasts with that for arithmetic in the set
{m + n
√
−5 : m, n ∈ Z},
where one has, for example,
6 = 2 × 3 = (1 +
√
−5) × (1 −
√
−5),](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-15-320.jpg)
![Prime number theory and the Riemann zeta-function 5
Conjecture 3. Let κ 2 be any constant. Then if H = (log N)κ
we should
have
π(N + H) − π(N) ∼
H
log N
as N → ∞.
This is supported by the following result due to Selberg in 1943 [15].
Theorem 4. Let f(N) be any increasing function for which f(N) → ∞ as
N → ∞. Assume the Riemann Hypothesis. Then there is a subset E of the
integers N, with
#{n ∈ E : n ≤ N} = o(N)
as N → ∞, such that
π(n + f(n) log2
n) − π(n) ∼ f(n) log n
for all n ∈ E.
Conjecture 3 would say that one can take E = ∅ if f(N) is a positive power
of log N.
Since Cramér’s Model leads inexorably to Conjecture 3, it came as quite a
shock to prime number theorists when the conjecture was disproved by Maier
[9] in 1985. Maier established the following result.
Theorem 5. For any κ 1 there is a constant δκ 0 such that
lim sup
N→∞
π(N + (log N)κ
) − π(N)
(log N)κ−1
≥ 1 + δκ
and
lim inf
N→∞
π(N + (log N)κ
) − π(N)
(log N)κ−1
≤ 1 − δκ.
The values of N produced by Maier, where π(N + (log N)κ
) − π(N) is
abnormally large, (or abnormally small), are very rare. None the less their
existence shows that the Cramér Model breaks down. Broadly speaking one
could summarize the reason for this failure by saying that arithmetic effects
play a bigger rôle than previously supposed. As yet we have no good alternative
to the Cramér model.
2 Open Questions About Primes,
and Important Results
Here are a few of the well-known unsolved problems about the primes.
(1) Are there infinitely many “prime twins” n, n+2 both of which are prime?
(Conjecture 2 gives a prediction for the rate at which the number of such
pairs grows.)](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-19-320.jpg)
![6 D. R. Heath-Brown
(2) Is every even integer n ≥ 4 the sum of two primes? (Goldbach’s Conjec-
ture.)
(3) Are there infinitely many primes of the form p = n2
+ 1?
(4) Are there infinitely many “Mersenne primes” of the form p = 2n
− 1?
(5) Are there arbitrarily long arithmetic progressions, all of whose terms are
prime?
(6) Is there always a prime between any two successive squares?
However there have been some significant results proved too. Here are a
selection.
(1) There are infinitely many primes of the form a2
+ b4
. (Friedlander and
Iwaniec [4], 1998.)
(2) There are infinitely many primes p for which p + 2 is either prime or a
product of two primes. (Chen [2], 1966.)
(3) There is a number n0 such that any even number n ≥ n0 can be written
as n = p+p
with p prime and p
either prime or a product of two primes.
(Chen [2], 1966.)
(4) There are infinitely many integers n such that n2
+ 1 is either prime or
a product of two primes. (Iwaniec [8], 1978.)
(5) For any constant c 243
205
= 1.185 . . ., there are infinitely many integers
n such that [nc
] is prime. Here [x] denotes the integral part of x, that is
to say the largest integer N satisfying N ≤ x. (Rivat and Wu [14], 2001,
after Piatetski-Shapiro, [11], 1953.)
(6) Apart from a finite number of exceptions, there is always a prime between
any two consecutive cubes. (Ingham [6], 1937.)
(7) There is a number n0 such that for every n ≥ n0 there is at least one
prime in the interval [n , n + n0.525
]. (Baker, Harman and Pintz, [1],
2001.)
(8) There are infinitely many consecutive primes p,
p such that p
− p ≤
(log p)/4. (Maier [10], 1988.)
(9) There is a positive constant c such that there are infinitely many consec-
utive primes p,
p such that
p
− p ≥ c log p
(log log p)(log log log log p)
(log log log p)2
.
(Rankin [13], 1938.)](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-20-320.jpg)
![Prime number theory and the Riemann zeta-function 7
(10) For any positive integer q and any integer a in the range 0 ≤ a q,
which is coprime to q, there are arbitrarily long strings of consecutive
primes, all of which leave remainder a on division by q. (Shiu [16], 2000.)
By way of explanation we should say the following. The result (1) demon-
strates that even though we cannot yet handle primes of the form n2
+ 1, we
can say something about the relatively sparse polynomial sequence a2
+ b4
.
The result in (5) can be viewed in the same context. One can think of [nc
] as
being a “polynomial of degree c” with c 1. Numbers (2), (3) and (4) are
approximations to, respectively, the prime twins problem, Goldbach’s prob-
lem, and the problem of primes of the shape n2
+ 1. The theorems in (6)
and (7) are approximations to the conjecture that there should be a prime
between consecutive squares. Of these (7) is stronger, if less elegant. Maier’s
result (8) shows that the difference between consecutive primes is sometimes
smaller than average by a factor 1/4, the average spacing being log p by the
Prime Number Theorem. (Of course the twin prime conjecture would be a
much stronger result, with differences between consecutive primes sometimes
being as small as 2.) Similarly, Rankin’s result (9) demonstrates that the
gaps between consecutive primes can sometimes be larger than average, by a
factor which is almost log log p. Again this is some way from what we expect,
since Conjecture 1 predict gaps as large as (log p)2
. Finally, Shiu’s result (10)
is best understood by taking q = 107
and a = 7, 777, 777, say. Thus a prime
leaves remainder a when divided by q, precisely when its decimal expansion
ends in 7 consecutive 7’s. Then (10) tells us that a table of primes will some-
where contain a million consecutive entries, each of which ends in the digits
7,777,777.
3 The Riemann Zeta-Function
In the theory of the zeta-function it is customary to use the variable s =
σ + it ∈ C. One then defines the complex exponential
n−s
:= exp(−s log n), with log n ∈ R.
The Riemann Zeta-function is then
ζ(s) :=
∞
n=1
n−s
, σ 1. (3.1)
The sum is absolutely convergent for σ 1, and for fixed δ 0 it is uniformly
convergent for σ ≥ 1 + δ. It follows that ζ(s) is holomorphic for σ 1. The
function is connected to the primes as follows.
Theorem 6. (The Euler Product.) If σ 1 then we have
ζ(s) =
p
(1 − p−s
)−1
,
where p runs over all primes, and the product is absolutely convergent.](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-21-320.jpg)
![Prime number theory and the Riemann zeta-function 9
4 The Analytic Continuation and Functional
Equation of ζ(s)
Our definition only gives a meaning to ζ(s) when σ 1. We now seek to
extend the definition to all s ∈ C. The key tool is the Poisson Summation
Formula .
Theorem 7. (The Poisson Summation Formula.) Suppose that f : R →
R is twice differentiable and that f, f
and f
are all integrable over R. Define
the Fourier transform by
ˆ
f(t) :=
∞
−∞
f(x)e−2πitx
dx.
Then ∞
−∞
f(n) =
∞
−∞
ˆ
f(n),
both sides converging absolutely.
There are weaker conditions under which this holds, but the above more
than suffices for our application. The reader should note that there are a
number of conventions in use for defining the Fourier transform, but the one
used here is the most appropriate for number theoretic purposes.
The proof (see Rademacher [12, page 71], for example) uses harmonic anal-
ysis on R+
. Thus it depends only on the additive structure and not on the
multiplicative structure.
If we apply the theorem to f(x) = exp{−x2
πv}, which certainly fulfils the
conditions, we have
ˆ
f(n) =
∞
−∞
e−x2 πv
e−2πinx
dx
=
∞
−∞
e−πv(x+in/v)2
e−πn2 /v
dx
= e−πn2 /v
∞
−∞
e−πvy2
dy
=
1
√
v
e−πn2 /v
,
providing that v is real and positive. Thus if we define
θ(v) :=
∞
−∞
exp(−πn2
v),
then the Poisson Summation Formula leads to the transformation formula
θ(v) =
1
√
v
θ(1/v).](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-23-320.jpg)
![12 D. R. Heath-Brown
6 The Product Formula
There is a useful product formula for ξ(s), due to Hadamard. In general we
have the following result, for which see Davenport [3, Chapter 11] for example.
Theorem 9. Let f(z) be an entire function with f(0) = 0, and suppose that
there are constants A 0 and θ 2 such that f(z) = O(exp(A|z|θ
)) for all
complex z. Then there are constants α and β such that
f(z) = eα+βz
∞
n=1
{(1 −
z
zn
)ez/zn
},
where zn runs over the zeros of f(z) counted with multiplicity. The infinite sum
∞
n=1 |zn|−2
converges, so that the product above is absolutely and uniformly
convergent in any compact set which includes none of the zeros.
We can apply this to ξ(s), since it is apparent from Theorem 8, together
with the definition (5.1) that
ξ(0) = ξ(1) =
1
2
π−1/2
Γ(
1
2
)Res{ζ(s); s = 1} =
1
2
.
For σ ≥ 2 one has ζ(s) = O(1) directly from the series (3.1), while Stir-
ling’s approximation yields Γ(s/2) = O(exp(|s| log |s|)). It follows that ξ(s) =
O(exp(|s| log |s|)) whenever σ ≥ 2. Moreover, when 1
2
≤ σ ≤ 2 one sees from
Theorem 8 that ξ(s) is bounded. Thus, using the functional equation, we can
deduce that ξ(s) = O(exp(|s| log |s|)) for all s with |s| ≥ 2.
We may therefore deduce from Theorem 9 that
ξ(s) = eα+βs
ρ
{(1 −
s
ρ
)es/ρ
},
where ρ runs over the zeros of ξ(s). Thus, with appropriate branches of the
logarithms, we have
log ξ(s) = α + βs +
ρ
{log(1 −
s
ρ
) +
s
ρ
}.
We can then differentiate termwise to deduce that
ξ
ξ
(s) = β +
ρ
{
1
s − ρ
+
1
ρ
},
the termwise differentiation being justified by the local uniform convergence
of the resulting sum. We therefore deduce that
ζ
ζ
(s) = β −
1
s − 1
+
1
2
log π −
1
2
Γ
Γ
(
s
2
+ 1) +
ρ
{
1
s − ρ
+
1
ρ
}, (6.1)
where, as ever, ρ runs over the zeros of ξ counted according to multiplicity. In
fact, on taking s → 1, one can show that
β = −
1
2
γ − 1 −
1
2
log 4π,
where γ is Euler’s constant. However we shall make no use of this fact.](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-26-320.jpg)
![Prime number theory and the Riemann zeta-function 23
10 Explicit Formulae
In this section we shall argue somewhat informally, and present results without
proof.
If f : (0, ∞) → C we define the Mellin transform of f to be the function
F(s) :=
∞
0
f(x)xs−1
dx.
By a suitable change of variables one sees that this is essentially a form of
Fourier transform. Indeed all the properties of Mellin transforms can readily
be translated from standard results about Fourier transforms. In particular,
under suitable conditions one has an inversion formula
f(x) =
1
2πi
σ+i∞
σ−i∞
F(s)x−s
ds.
Arguing purely formally one then has
∞
n=2
Λ(n)f(n) =
∞
n=2
Λ(n)
1
2πi
2+i∞
2−i∞
F(s)n−s
ds
=
1
2πi
2+i∞
2−i∞
{−
ζ
ζ
(s)}F(s)ds.
If one now moves the line of integration to (s) = −N one passes poles at
s = 1 and at s = ρ for every non-trivial zero ρ, as well as at the trivial zeros
−2n. Under suitable conditions the integral along (s) = −N will tend to 0
as N → ∞. This argument leads to the following result.
Theorem 17. Suppose that f ∈ C2
(0, ∞) and that supp(f) ⊆ [1, X] for some
X. Then
∞
n=2
Λ(n)f(n) = F(1) −
ρ
F(ρ) −
∞
n=1
F(−2n).
One can prove such results subject to weaker conditions on f. If x is given,
and
f(t) =
1, t ≤ x
0, t x,
then the conditions above are certainly not satisfied, but we have the following
related result.
Theorem 18. (The Explicit Formula.) Let x ≥ T ≥ 2. Then
ψ(x) = x −
ρ: |γ|≤T
xρ
ρ
+ O(
x log2
x
T
).](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-37-320.jpg)
![24 D. R. Heath-Brown
For a proof of this see Davenport [3, Chapter 17], for example. There are
variants of this result containing a sum over all zeros, and with no error term,
but the above is usually more useful.
The explicit formula shows exactly how the zeros influence the behaviour
of ψ(x), and hence of π(x). The connection between zeros and primes is even
more clearly shown by the following result of Landau.
Theorem 19. For fixed positive real x define Λ(x) = 0 if x ∈ N and Λ(x) =
Λ(n) if x = n ∈ N. Then
Λ(x) = −
2π
T
ρ: 0γ≤T
xρ
+ Ox(
log T
T
),
where Ox(. . .) indicates that the implied constant may depend on x.
This result shows that the zeros precisely determine the primes. Thus, for
example, one can reformulate the conjecture (1.1) as a statement about the
zeros of the zeta-function. All the unevenness of the primes, for example the
behaviour described by Theorem 5, is encoded in the zeros of the zeta-function.
It therefore seems reasonable to expect that the zeros themselves should have
corresponding unevenness.
11 Dirichlet Characters
We now turn to the simplest type of generalization of the Riemann Zeta-
function, namely the Dirichlet L-functions. In the remainder of these notes we
shall omit most of the proofs, being content merely to describe what can be
proved.
A straightforward example of a Dirichlet L-function is provided by the
infinite series
1 −
1
3s
+
1
5s
−
1
7s
+
1
9s
−
1
11s
+ . . . . (11.1)
We first need to describe the coefficients which arise.
Definition . Let q ∈ N. A “(Dirichlet) character χ to modulus q” is a function
χ : Z → C such that
(i) χ(mn) = χ(m)χ(n) for all m, n ∈ Z;
(ii) χ(n) has period q;
(iii) χ(n) = 0 whenever (n, q) = 1; and
(iv) χ(1) = 1.](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-38-320.jpg)
![Prime number theory and the Riemann zeta-function 25
Part (iv) of the definition is necessary merely to rule out the possibility
that χ is identically zero.
As an example we can take the function
χ(n) =
⎧
⎨
⎩
1, n ≡ 1 (mod 4),
−1, n ≡ 3 (mod 4),
0, n ≡ 0 (mod 2).
(11.2)
This is a character modulo 4, and is the one generating the series (11.1). A
second example is the function
χ0(n) :=
1, (n, q) = 1,
0, (n, q) = 1.
This produces a character for every modulus q, known as the principal char-
acter modulo q.
A number of key facts are gathered together in the following theorem.
Theorem 20. (i) We have |χ(n)| = 1 for every n coprime to q.
(ii) If χ1 and χ2 are two characters to modulus q, then so is χ1χ2, where we
define χ1χ2(n) = χ1(n)χ2(n).
(iii) There are exactly ϕ(q) different characters to modulus q.
(iv) If n ≡ 1(mod q) then there is at least one character χ modulo q for which
χ(n) = 1.
In part (iii) the function ϕ(q) is the number of positive integers n ≤ q for
which n and q are coprime.
To prove part (i) we note that the sequence nk
(mod q) must eventually
repeat when k runs through N. Thus there exist k j with χ(nk
) = χ(nj
),
and hence χ(n)k
= χ(n)j
. Since n is coprime to q we have χ(n) = 0, so that
χ(n)j−k
= 1.
Part (ii) of the theorem is obvious, but parts (iii) and (iv) are harder, and
we refer the reader to Davenport [3, Chapter 4] for the details. As an example
of (iii) we note that ϕ(4) = 2, and we have already found two characters
modulo 4. There are no others.
One further fact may elucidate the situation. Consider a general finite
abelian group G. In our case we will have G = (Z/qZ)×
. Thus G will consist
of those residue classes n mod q for which (n, q) = 1, with the multiplica-
tion operation. Define
G to be the group of homomorphisms θ : G → C×
,
where the group action is given by (θ1θ2)(g) := θ1(g)θ2(g). In our case these
homomorphisms are, in effect, the characters. Then the groups G and
G are
isomorphic, and part (iii) above is an immediate consequence. The details can
be found in Ireland and Rosen [7, pages 253 and 254], for example. Indeed](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-39-320.jpg)
![Prime number theory and the Riemann zeta-function 27
For example, we may take χ to be the character modulo 4 given by (11.2).
Then if q = 12 we induce a character ψ modulo 12, as in the following table.
1 2 3 4 5 6 7 8 9 10 11 12
χ 1 0 -1 0 1 0 -1 0 1 0 -1 0
ψ 1 0 0 0 1 0 -1 0 0 0 -1 0
A character χ(mod q) which cannot be produced this way from some divisor
r q is said to be “primitive”. The principal character is induced by the
character χ0
(mod 1), that is to say by the character which is identically 1.
If q is prime, then all the characters except for the principal character are
primitive. In general there will be both primitive and imprimitive characters
to each modulus. Imprimitive characters are a real nuisance!!
12 Dirichlet L-functions
For any character χ to modulus q we will define the corresponding Dirichlet
L-function by setting
L(s, χ) =
∞
n=1
χ(n)
ns
, (σ 1).
We content ourselves here with describing the key features of these func-
tions, and refer the reader to Davenport [3], for example, for details.
The sum is absolutely convergent for σ 1 and is locally uniformly con-
vergent, so that L(s, χ) is holomorphic in this region. If χ is the principal
character modulo q then the series fails to converge when σ ≤ 1. However
for non-principal χ the series is conditionally convergent when σ 0, and the
series defines a holomorphic function in this larger region.
There is an Euler product identity
L(s, χ) =
p
(1 − χ(p)p−s
)−1
, (σ 1).
This can be proved in the same way as for ζ(s) using the multiplicativity of
the function χ.
Suppose that χ is primitive, and that χ(−1) = 1. If we apply the Poisson
summation formula to
f(x) = e−(a+qx)2 πv/q
,
multiply the result by χ(a), and sum for 1 ≤ a ≤ q, we find that
θ(v, χ) =
τ(χ)
√
q
1
√
v
θ(
1
v
, χ),
where
θ(v, χ) :=
∞
n=−∞
χ(n)e−n2 πv/q](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-41-320.jpg)
![Prime number theory and the Riemann zeta-function 29
for T ≥ 2, which can be seen as an analogue of the Riemann – von Mangoldt
formula. This shows in particular that the interval [T, T + 1] contains around
1
2π
log
qT
2π
zeros, on average.
The work on regions without zeros can be generalized, but there are serious
problems with possible zeros on the real axis. Thus one can show that there
is a constant c 0, which is independent of q, such that if T ≥ 2 then L(s, χ)
has no zeros in the region
σ ≥ 1 −
c
log qT
, 0 |t| ≤ T.
If χ is not a real-valued character then we can extend this result to the case
t = 0, but is a significant open problem to deal with the case in which χ is real.
However in many other important aspects techniques used for the Riemann
Zeta-function can be successfully generalized to handle Dirichlet L-functions.
References
[1] R.C. Baker, G. Harman, and J. Pintz, The difference between consecutive
primes. II., Proc. London Math. Soc. (3), 83 (2001), 532-562.
[2] J.-R. Chen, on the representation of a large even integer as a sum of a
prime and a product of at most two primes, Kexue Tongbao, 17 (1966),
385-386.
[3] H. Davenport, Multiplicative number theory, Graduate Texts in Mathe-
matics, 74. (Springer-Verlag, New York-Berlin, 1980).
[4] J.B. Friedlander and H. Iwaniec, The polynomial X2
+ Y 4
captures its
primes, Annals of Math. (2), 148 (1998), 945-1040.
[5] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers,
(Oxford University Press, New York, 1979).
[6] A.E. Ingham, On the difference between consecutive primes, Quart. J.
Math. Oxford, 8 (1937), 255-266.
[7] K. Ireland and M. Rosen, A classical introduction to modern number the-
ory, Graduate Texts in Math., 84, (Springer, Heidelberg-New York, 1990).
[8] H. Iwaniec, Almost-primes represented by quadratic polynomials, Invent.
Math., 47 (1978), 171-188.
[9] H. Maier, Primes in short intervals, Michigan Math. J., 32 (1985), 221-
225.](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-43-320.jpg)
![30 D. R. Heath-Brown
[10] H. Maier, Small differences between prime numbers, Michigan Math. J.,
35 (1988), 323-344.
[11] I.I. Piatetski-Shapiro, On the distribution of prime numbers in sequences
of the form [f(n)], Mat. Sbornik N.S., 33(75) (1953), 559-566.
[12] H. Rademacher, Topics in analytic number theory, Grundlehren math.
Wiss., 169, (Springer, New York-Heidelberg, 1973).
[13] R.A. Rankin, The difference between consecutive prime numbers, J. Lon-
don Math. Soc., 13 (1938), 242-247.
[14] J. Rivat and J. Wu, Prime numbers of the form [nc
], Glasg. Math. J., 43
(2001), 237-254.
[15] A. Selberg, On the normal density of primes in small intervals, and the
difference between consecutive primes, Arch. Math. Naturvid., 47, (1943),
87-105.
[16] D.K.L. Shiu, Strings of congruent primes, J. London Math. Soc. (2), 61
(2000), 359-373.
Mathematical Institute,
24-29, St. Giles’,
Oxford OX1 3LB
rhb@maths.ox.ac.uk](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-44-320.jpg)
![32 Yan V. Fyodorov
they belong both to the family of invariant ensembles, and to the family of
ensembles with independent, identically distributed (i.i.d) entries. In general,
mathematical methods used to treat those two families are very different.
In fact, all random matrix techniques and ideas can be most clearly and
consistently introduced using Gaussian case as a paradigmatic example. In the
present set of lectures we mainly concentrate on consequences of the invari-
ance of the corresponding probability density function, leaving aside methods
of exploiting statistical independence of matrix entries. Under these circum-
stances the method of orthogonal polynomials is the most adequate one, and
for the Gaussian case the relevant polynomials are Hermite polynomials. Being
mostly interested in the limit of large matrix sizes we will spend a considerable
amount of time investigating various asymptotic regimes of Hermite polyno-
mials, since the latter are main building blocks of various correlation functions
of interest. In the last part of our lecture course we will discuss why statistics
of characteristic polynomials of random Hermitian matrices turns out to be
interesting and informative to investigate, and will make a contact with recent
results in the domain.
The presentation is quite informal in the sense that I will not try to prove
various statements in full rigor or generality. I rather attempt outlining the
main concepts, ideas and techniques preferring a good illuminating example
to a general proof. A much more rigorous and detailed exposition can be
found in the cited literature. I will also frequently employ the symbol ∝. In
the present set of lectures it always means that the expression following ∝
contains a multiplicative constant which is of secondary importance for our
goals and can be restored when necessary.
2 Introduction
In these lectures we use the symbol T
to denote matrix or vector transposition
and the asterisk ∗
to denote Hermitian conjugation. In the present section the
bar z denotes complex conjugation.
Let us start with a square complex matrix Ẑ of dimensions N × N, with
complex entries zij = xij + iyij, 1 ≤ i, j ≤ N. Every such matrix can be
conveniently looked at as a point in a 2N2
-dimensional Euclidean space with
real Cartesian coordinates xij, yij, and the length element in this space is
defined in a standard way as:
(ds)2
= Tr
dẐdẐ∗
=
ij
dzijdzij =
ij
(dx)2
ij + (dy)2
ij
. (2.1)
As is well-known (see e.g.[1]) any surface embedded in an Euclidean space
inherits a natural Riemannian metric from the underlying Euclidean struc-
ture. Namely, let the coordinates in a n−dimensional Euclidean space be
(x1, . . . , xn), and let a k−dimensional surface embedded in this space be param-
eterized in terms of coordinates (q1, . . . , qk), k ≤ n as xi = xi(q1, . . . , qk), i =](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-46-320.jpg)
![34 Yan V. Fyodorov
which results in (ds)2
= R2
(dθ)2
+ R2
sin2
θ(dφ)2
. Hence the matrix elements
of the metric are g11 = R2
, g12 = g21 = 0, g22 = R2
sin2
θ, and the cor-
responding “volume element” on the sphere is the familiar elementary area
dµ = R2
sin θdθdφ.
As a less trivial example to be used later on consider a 2−dimensional
manifold formed by 2 × 2 unitary matrices Û embedded in the 8 dimensional
Euclidean space of Gl(2; C) matrices. Every such matrix can be represented
as the product of a matrix Ûc from the coset space U(2)/U(1) × U(1) param-
eterized by k = 2 real coordinates 0 ≤ φ 2π, 0 ≤ θ ≤ π/2, and a diagonal
unitary matrix Ud, that is Û = ÛdÛc, where
Ûc =
cos θ − sin θe−iφ
sin θeiφ
cos θ
, ˆ
Ud =
e−iφ1
0
0 eiφ2
. (2.4)
Then the differential dÛ of the matrix Û = ÛdÛc has the following form:
ˆ
dU =
−[dθ sin θ + i cos θdφ1 ]e−iφ1 e−i(φ1 + φ) [−dθ cos θ + i(dφ1 + dφ) sin θ]
ei(φ+ φ2 ) [dθ cos θ + i(dφ + dφ2 ) sin θ] [−dθ sin θ + idφ2 cos θ]eiφ2
, (2.5)
which yields the length element and the induced Riemannian metric:
(ds)2
= Tr
dÛdÛ∗
(2.6)
= 2(dθ)2
+ (dφ1)2
+ (dφ2)2
+ 2 sin2
θ(dφ)2
+2 sin2
θ(dφ dφ1 + dφ dφ2).
We see that the nonzero entries of the Riemannian metric tensor gmn in this
case are g11 = 2, g22 = g33 = 1, g44 = 2 sin2
θ, g24 = g42 = g34 = g43 =
sin2
θ, so that the determinant det [gmn] = 4 sin2
θ cos2
θ. Finally, the induced
integration measure on the group U(2) is given by
dµ(Û) = 2 sin θ cos θ dθ dφ dφ1 dφ2. (2.7)
It is immediately clear that the above expression is invariant, by construction,
with respect to multiplications Û → V̂ Û, for any fixed unitary matrix V from
the same group. Therefore, Eq.(2.7) is just the Haar measure on the group.
We will make use of these ideas several times in our lectures. Let us now
concentrate on the N2
−dimensional subspace of Hermitian matrices in the
2N2
− dimensional space of all complex matrices of a given size N. The Her-
miticity condition Ĥ = Ĥ∗
≡ ĤT amounts to imposing the following restric-
tions on the coordinates: xij = xji, yij = −yji. Such a restriction from the
space of general complex matrices results in the length and volume element on
the subspace of Hermitian matrices:
(ds)2
= Tr
dĤdĤ∗
=
i
(dxii)2
+ 2
ij
(dxij)2
+ (dyij)2
(2.8)
dµ(Ĥ) = 2
N (N −1)
2
i
dxii
ij
dxijdyij. (2.9)](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-48-320.jpg)
![Introduction to the random matrix theory: Gaussian Unitary
Ensemble and beyond
35
Obviously, the length element (ds)2
= TrdĤdĤ∗
is invariant with respect to
an automorphism (a mapping of the space of Hermitian matrices to itself) by a
similarity transformation Ĥ → U−1
ĤÛ, where Û ∈ U(N) is any given unitary
N × N matrix: Û∗
= Û−1
. Therefore the corresponding integration measure
dµ(Ĥ) is also invariant with respect to all such “rotations of the basis”.
The above-given measure dµ(Ĥ) written in the coordinates xii, xij, yij is
frequently referred to as the “flat measure”. Let us discuss now another, very
important coordinate system in the space of Hermitian matrices which will be
in the heart of all subsequent discussions. As is well-known, every Hermitian
matrix Ĥ can be represented as
Ĥ = ÛΛ̂Û−1
, Λ̂ = diag(λ1, . . . , λN ), Û∗
Û = ˆ
I, (2.10)
where real −∞ λk ∞, k = 1, . . . , N are eigenvalues of the Hermitian
matrix, and rows of the unitary matrix Û are corresponding eigenvectors.
Generically, we can consider all eigenvalues to be simple (non-degenerate).
More precisely, the set of matrices Ĥ with non-degenerate eigenvalues is dense
and open in the N2
-dimensional space of all Hermitian matrices, and has
full measure (see [3], p.94 for a formal proof). The correspondence Ĥ →
Û ∈ U(N), Λ̂
is, however, not one-to-one, namely Û1Λ̂Û−1
1 = Û2Λ̂Û−1
2 if
Û−1
1 Û2 = diag
eiφ1
, . . . , eiφN
for any choice of the phases φ1, . . . , φN . To
make the correspondence one-to-one we therefore have to restrict the unitary
matrices to the coset space U(N)/U(1) ⊗ . . . ⊗ U(1), and also to order the
eigenvalues, e.g. requiring λ1 λ2 . . . λN . Our next task is to write
the integration measure dµ(Ĥ) in terms of eigenvalues Λ̂ and matrices Û. To
this end, we differentiate the spectral decomposition Ĥ = ÛΛ̂Û∗
, and further
exploit: d
Û∗
Û
= dÛ∗
Û + Û∗
dÛ = 0. This leads to
dĤ = Û
dΛ̂ + Û∗
dÛΛ̂ − Λ̂Û∗
dÛ
Û∗
. (2.11)
Substituting this expression into the length element (ds)2
, see Eq.(2.8), and us-
ing the short-hand notation δÛ for the matrix Û∗
dÛ satisfying anti-Hermiticity
condition δÛ∗
= −δÛ, we arrive at:
(ds)2
= Tr
dΛ̂
2
+ 2dΛ̂
δÛΛ̂ − Λ̂δÛ
(2.12)
+
δÛΛ̂
2
+
Λ̂δÛ
2
− 2δÛΛ̂2
δÛ .
Taking into account that Λ̂ is purely diagonal, and therefore the diagonal
entries of the commutator
δÛΛ̂ − Λ̂δÛ
are zero, we see that the second
term in Eq.(2.12) vanishes. On the other hand, the third and subsequent](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-49-320.jpg)
![36 Yan V. Fyodorov
terms when added up are equal to
2Tr
δÛΛ̂δÛΛ̂ − δÛ2
Λ̂2
= 2
ij
δUijλjδUjiλi − λ2
i δUijδUji
= −
ij
(λi − λj)2
δUjiδUij
which together with the first term yields the final expression for the length
element in the “spectral” coordinates
(ds)2
=
i
(dλi)2
+
ij
(λi − λj)2
δUijδUij (2.13)
where we exploited the anti-Hermiticity condition −δUji = δUij. Introducing
the real and imaginary parts δUij = δpij +iδqij as independent coordinates we
can calculate the corresponding integration measure dµ(Ĥ) according to the
rule in Eq.(2.3), to see that it is given by
dµ(Ĥ) =
ij
(λi − λj)2
i
dλi × dM(Û) . (2.14)
The last factor dM(U) stands for the part of the measure depending only on
the U−variables. A more detailed consideration shows that, in fact, dM(Û) ≡
dµ(Û), which means that it is given (up to a constant factor) by the invariant
Haar measure on the unitary group U(N). This fact is however of secondary
importance for the goals of the present lecture.
Having an integration measure at our disposal, we can introduce a probabil-
ity density function (p.d.f.) P(Ĥ) on the space of Hermitian matrices, so that
P(Ĥ)dµ(Ĥ) is the probability that a matrix Ĥ belongs to the volume element
dµ(Ĥ). Then it seems natural to require for such a probability to be invariant
with respect to all the above automorphisms, i.e. P(Ĥ) = P
Û∗
ĤÛ
. It
is easy to understand that this “postulate of invariance” results in P being a
function of N first traces TrĤn
, n = 1, . . . , N (the knowledge of first N traces
fixes the coefficients of the characteristic polynomial of Ĥ uniquely, and hence
the eigenvalues. Therefore traces of higher order can always be expressed in
terms of the lower ones). Of particular interest is the case
P(Ĥ) = C exp −Tr Q(Ĥ), Q(x) = a2jx2j
+ . . . + a0, (2.15)
where 2j ≤ N, the parameters a2l and C are real constants, and a2j 0.
Observe that if we take
Q(x) = ax2
+ bx + c, (2.16)
then e−T r Q(Ĥ)
takes the form of the product
e−a[
i x2
ii +2
i j (x2
ij +y2
ij )]e−b
i xii
e−cN
(2.17)
= e−cN
N
i=1
e−ax2
ii −bxii
ij
e−2ax2
ij
ij
e−2ay2
ij .](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-50-320.jpg)
![Introduction to the random matrix theory: Gaussian Unitary
Ensemble and beyond
37
We therefore see that the probability distribution of the matrix Ĥ can be rep-
resented as a product of factors, each factor being a suitable Gaussian distribu-
tion depending only on one variable in the set of all coordinates xii, xij, yij.
Since the same factorization is valid also for the integration measure dµ(Ĥ), see
Eq.(2.9), we conclude that all these N2
variables are statistically independent
and Gaussian-distributed.
A much less obvious statement is that if we impose simultaneously two
requirements:
• The probability density function P(Ĥ) is invariant with respect to all
conjugations Ĥ → Ĥ
= U−1
ĤÛ by unitary matrices Û, that is P(Ĥ
) =
P(Ĥ); and
• the N2
variables xii, xij, yij are statistically independent, i.e.
P(Ĥ) =
N
i=1
fi(xii)
N
ij
f
(1)
ij (xij)f
(2)
ij (yij), (2.18)
then the function P(Ĥ) is necessarily of the form P(Ĥ) = Ce−(aTrĤ2 +bTrĤ+cN),
for some constants a 0, b, c. The proof for any N can be found in [2], and
here we just illustrate its main ideas for the simplest, yet nontrivial case N = 2.
We require invariance of the distribution with respect to the conjugation of
Ĥ by Û ∈ U(2), and first consider a particular choice of the unitary matrix
Û =
1 −θ
θ 1
corresponding to φ = φ1 = φ2 = 0, and small values θ 1
in Eq.(2.4). In this approximation the condition Ĥ
= U−1
ĤÛ amounts to
x
11 x
12 + iy
12
x
12 − iy
12 x
22
(2.19)
=
x11 + 2θx12 x12 + iy12 + θ (x22 − x11)
x12 − iy12 + θ (x22 − x11) x22 − 2θx12
,
where we kept only terms linear in θ. With the same precision we expand the
factors in Eq.(2.18):
f1(x
1) = f1(x1) 1 + 2θx12
1
f1
df1
dx11
, f2(x
22) = f2(x22) 1 − 2θx12
1
f2
df2
dx22
f
(1)
12 (x
12) = f
(1)
12 (x12)
1 + θ(x22 − x11)
1
f
(1)
12
df
(1)
12
dx12
, f
(2)
12 (y
21) = f
(2)
12 (y12).
The requirements of statistical independence and invariance amount to the
product of the left-hand sides of the above expressions to be equal to the
product of the right-hand sides, for any θ. This is possible only if:
2x12
d ln f1
dx11
−
d ln f2
dx22
+ (x22 − x11)
d ln f
(1)
12
dx12
= 0, (2.20)](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-51-320.jpg)
![38 Yan V. Fyodorov
which can be further rewritten as
1
(x22 − x11)
d ln f1
dx11
−
d ln f2
dx22
= const =
1
2x12
d ln f
(1)
12
dx12
, (2.21)
where we used that the two sides in the equation above depend on essentially
different sets of variables. Denoting const1 = −2a, we see immediately that
f
(1)
12 (x12) ∝ e−2ax2
12 ,
and further notice that
d ln f1
dx11
+ 2ax11 = const2 =
d ln f2
dx22
+ 2ax22
by the same reasoning. Denoting const2 = −b, we find:
f1(x11) ∝ e−ax2
11 −bx11
, f2(x22) ∝ e−ax2
22 −bx22
, (2.22)
and thus we are able to reproduce the first two factors in Eq.(2.17). To repro-
duce the remaining factors we consider the conjugation by the unitary matrix
ˆ
Ud =
1 − iα 0
0 1 + iα
, which corresponds to the choice θ = 0, φ1 = φ2 =
−α = in Eq.(2.4), and again we keep only terms linear in the small parameter
α 1. Within such a precision the transformation leaves the diagonal entries
x11, x22 unchanged, whereas the real and imaginary parts of the off-diagonal
entries are transformed as
x
12 = x12 − 2αy12, y
12 = y12 + 2αx12.
In this case the invariance of the p.d.f. P(Ĥ) together with the statistical
independence of the entries amount, after straightforward manipulations, to
the condition
1
x12
d ln f
(1)
12
dx12
=
1
y12
d ln f
(2)
12
dy12
which together with the previously found f
(1)
12 (x12) yields
f
(2)
12 (y12) ∝ e−2ay2
12 ,
completing the proof of Eq.(2.17).
The Gaussian form of the probability density function, Eq.(2.17), can also
be found as a result of rather different lines of thought. For example, one
may invoke an information theory approach a la Shanon-Khinchin and define
the amount of information I[P(Ĥ)] associated with any probability density
function P(Ĥ) by
I[P(Ĥ)] = −
dµ(Ĥ) P(Ĥ) ln P(Ĥ) (2.23)](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-52-320.jpg)
![Introduction to the random matrix theory: Gaussian Unitary
Ensemble and beyond
39
This is a natural extension of the corresponding definition I[p1, . . . , pm] =
−
m
l=1 pm ln pm for discrete events 1, ..., m.
Now one can argue that in order to have matrices Ĥ as random as possible
one has to find the p.d.f. minimizing the information associated with it for
a certain class of P(H) satisfying some conditions. The conditions usually
have a form of constraints ensuring that the probability density function has
desirable properties. Let us, for example, impose the only requirement that
the ensemble average for the two lowest traces TrĤ, TrĤ2
must be equal to
certain prescribed values, say E
TrĤ
= b and E
TrĤ2
= a 0, where
the E [. . .] stand for the expectation value with respect to the p.d.f. P(H).
Incorporating these constraints into the minimization procedure in a form of
Lagrange multipliers ν1, ν2, we seek to minimize the functional
I[P(Ĥ)] = −
dµ(Ĥ) P(Ĥ)
ln P(Ĥ) − ν1TrĤ − ν2TrĤ2
. (2.24)
The variation of such a functional with respect to δP(H) results in
δI[P(Ĥ)] = −
dµ(Ĥ) δP(Ĥ)
1 + ln P(Ĥ) − ν1TrĤ − ν2TrĤ2
= 0
(2.25)
possible only if
P(Ĥ) ∝ exp{ν1TrĤ + ν2TrĤ2
}
again giving the Gaussian form of the p.d.f. The values of the Lagrange mul-
tipliers are then uniquely fixed by constants a, b, and the normalization con-
dition on the probability density function. For more detailed discussion, and
for further reference see [2], p.68.
Finally, let us discuss yet another construction allowing one to arrive at
the Gaussian Ensembles exploiting the idea of Brownian motion. To start
with, consider a system whose state at time t is described by one real vari-
able x, evolving in time according to the simplest linear differential equation
d
dt
x = −x describing a simple exponential relaxation x(t) = x0e−t
towards
the stable equilibrium x = 0. Suppose now that the system is subject to a
random additive Gaussian white noise ξ(t) function of intensity D 1
, so that
the corresponding equation acquires the form
d
dt
x = −x + ξ(t), Eξ [ξ(t1)ξ(t1)] = Dδ(t1 − t2), (2.26)
1
The following informal but instructive definition of the white noise process may be
helpful for those not very familiar with theory of stochastic processes. For any 0 t 2π
and integer k ≥ 1 define the random function ξk (t) = 2/π
k
n=0 an cos nt, where real
coefficients an are all independent, Gaussian distributed with zero mean E[an ] = 0 and
variances E[a2
0] = D/2 and E[a2
n ] = D for 1 ≤ n ≤ k. Then one can, in a certain sense,
consider white noise as the limit of ξk (t) for k → ∞. In particular, the Dirac δ(t − t
) is
approximated by the limiting value of sin [(k+1/2)(t−t
)]
2π sin (t−t)/2](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-53-320.jpg)
![40 Yan V. Fyodorov
where Eξ[. . .] stands for the expectation value with respect to the random
noise. The main characteristic property of a Gaussian white noise process is
the following identity:
Eξ exp
b
a
ξ(t)v(t)dt = exp
D
2
b
a
v2
(t)dt (2.27)
valid for any (smooth enough) test function v(t). This is just a direct general-
ization of the standard Gaussian integral identity:
∞
−∞
dq
√
2πa
e− 1
2a
q2 +qb
= e
ab2
2 , (2.28)
valid for Re a 0, and any (also complex) parameter b.
For any given realization of the Gaussian random process ξ(t) the solution
of the stochastic differential equation Eq.(2.26) is obviously given by
x(t) = e−t
x0 +
t
0
eτ
ξ(τ)dτ . (2.29)
This is a random function, and our main goal is to find the probability density
function P(t, x) for the variable x(t) to take value x at any given moment in
time t, if we know surely that x(0) = x0. This p.d.f. can be easily found from
the characteristic function
F(t, q) = Eξ
e−iqx(t)
= exp
−iqx0e−t
−
Dq2
4
(1 − e−2t
) (2.30)
obtained by using Eqs. (2.27) and (2.29). The p.d.f. is immediately recovered
by employing the inverse Fourier transform:
P(t, x) =
∞
−∞
dq
2π
eiqx
Eξ
e−iqx(t)
(2.31)
=
1
πD(1 − e−2t)
exp
−
(x − x0e−t
)
2
D(1 − e−2t)
.
The formula Eq.(2.31) is called the Ornstein-Uhlenbeck (OU) probability
density function, and the function x(t) satisfying the equation Eq.(2.26) is
known as the O-U process. In fact, such a process describes an interplay
between the random “kicks” forcing the system to perform a kind of Brownian
motion and the relaxation towards x = 0. It is easy to see that when time
grows the OU p.d.f. “forgets” about the initial state and tends to a stationary
(i.e. time-independent) universal Gaussian distribution:
P(t → ∞, x) =
1
√
πD
exp
−
x2
D
. (2.32)](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-54-320.jpg)
![made no attempt to modify for these the provisions of the General
Consolidated Order of 1847, the effect of which upon the workhouse
administration of the period we have already described.[358]
Meanwhile the workhouse schools continued to improve very
slowly in educational efficiency. The policy of the Central Authority
was apparently to develop industrial training—agricultural work, the
simpler handicrafts, and domestic service—on the model of the
Quatt School in Shropshire. Whether or not this industrial work
militated against more intellectual accomplishments is a moot point,
but we hear of the reports of 'the stagnant dulness of workhouse
education' which annually proceed from Her Majesty's Inspectors of
Schools.[359]
Whether or not from a certain divergence of aim between the
departments, the connection was in 1863 severed,[360]
and the Poor
Law Board thenceforward had its own inspectors of Poor Law
Schools, whose criticisms and complaints, all in favour of the large
district schools as compared with the single union school, appear
from 1867 onward in the Annual Reports.[361]
At the very end of the period we may note the beginning of a
reaction against the barrack schools. It was pointed out by those
acquainted with the Scottish system of boarding-out, as well as by
persons experienced in English Poor Law administration, that these
expensive boarding schools were not answering so well as their
admirers claimed, especially as regards the girls. During 1866-9 the
alternative of boarding-out children in private families at 4s. a
week (now 5s.) was warmly discussed, and experimentally adopted
in a few places.[362]
In 1869 the Central Authority so far yielded to
the criticisms made upon these institutions as to permit, under
elaborate restrictions and safeguards, the boarding-out, in families
beyond the limits of the union, of the comparatively small class of
children who were actually or practically orphans.[363]
In these cases
all idea of making the condition of the pauper child less eligible than
that of the lowest independent labourer was definitely abandoned.](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-56-320.jpg)
![The whole concern of the Central Authority was to see that the
provision for the boarded-out child was good and complete. Far from
being assimilated to the children of the lowest independent
labourers, the boarded-out children were only to be entrusted to
specially selected families superior to the lowest, who undertook to
bring them up as their own, to provide proper food, clothing and
washing, to train them in good habits as well as in suitable domestic
and industrial work, and to make them regularly attend school and
place of worship. For all this the foster parents were to receive with
each child a sum three or four times as great as was, with the
sanction of the Central Authority, commonly allowed for the
maintenance of each of the couple of hundred thousand children at
that date on outdoor relief; and which (as Professor Fawcett vainly
objected) was far in excess of what the ordinary labourer could
afford to expend on his own children.[364]
A plan, observed Mr.
Fowle, which cannot be defended on any sound principles of Poor
Law.[365]
It is indeed impossible, says Mr. Mackay in this
connection, to deny that apparently every provision for pauper
children may be regarded as a contravention of this rule.... Professor
Fawcett's ... argument has been tacitly neglected.[366]
E.—The Sick
We have shown that, between 1834 and 1847, it was not
contemplated that persons actually sick would be received in the
workhouse, and that there was no trace in the documents of any
desire on the part of the Central Authority to interfere with the usual
practice of granting to them outdoor relief, which had not been in
any way condemned or discredited by the 1834 Report. The same
may be said of the Statutes, Orders, and Circulars of 1847-71. We
find no suggestion that the boards of guardians ought not to grant
outdoor relief in cases of sickness, or that sick paupers ought to be
relieved in the workhouse. On the contrary, the exceptions
specifically made in favour of sick persons seem to be even widened
in scope. Thus, in 1848, the Central Authority laid it down that
widows with illegitimate children were not to be refused outdoor](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-57-320.jpg)
![relief, if the children were sick.[367]
By the Outdoor Relief Regulation
Order of December 1852, it was definitely provided that outdoor
relief might be given in case of sickness in the family, even if the
head of the family was simultaneously earning wages.[368]
The same
policy was embodied in the corresponding General Order issued on
1st January 1869, to certain Metropolitan unions.[369]
Further, in the
panic about cholera in 1866, the Central Authority informed the
boards of guardians by circular that in cases of emergency they
might call in any medical and other assistance that was needed, and
even provide whatever sustenance, clothing, etc., was required,[370]
apparently irrespective of destitution and of all General Orders,
etc., to the contrary. Moreover, early in this period we note the
beginning of the special definition of destitution as regards medical
relief which has since been acted upon, that is to say, the inability to
pay for the medical attendance that the nature of the case requires.
Thus it was declared by the Central Authority in 1848 that the parish
doctor might attend sick servants living in their master's household,
who were plainly not destitute in the ordinary sense, as not being
without food and lodging, but who, if there were no wages due to
them, might be unable to pay for medical attendance.[371]
A similar
line of thought may be traced in that provision of the Act of 1851
which authorised boards of guardians to make annual subscriptions
out of the poor rate to public hospitals and infirmaries, to enable
these non-pauper institutions the better to provide for the poor.[372]
The sick wards of the workhouses, as the Central Authority
explained in 1869, were originally provided for the cases of paupers
in the workhouse who might be attacked by illness; and not as State
hospitals into which all the sick poor of the country might be
received for medical treatment and care. So far is this, indeed, from
being the case that at least two-thirds of the sick poor receive
medical attendance and treatment in their own homes.[373]
When in
1869-71, the Central Authority obtained elaborate reports showing,
for all parts of England, the practice that prevailed of normally giving
outdoor relief to the sick, and of taking them into the workhouse
infirmaries only when this was called for by (a) the nature of the](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-58-320.jpg)
![disease, (b) the wishes of the patient, or (c) the nature of the home,
and then only where suitable infirmary accommodation was
available, there is no indication that any objection was entertained to
the policy of outdoor relief to this large class.[374]
What is new in this period is the appearance, as a positive policy, of
bringing pressure to bear on the boards of guardians to improve the
quality of the medical attendance and medicine supplied. This led to
an explicit disavowal, so far as regards the sick paupers, of any
application to them of the principle of making the pauper's condition
less eligible than that of the lowest grade of independent labourers.
It is noteworthy that this new departure applied to outdoor medical
relief quite as much as to institutional medical treatment, in which it
has subsequently been sometimes excused on the ground that the
superior treatment is accompanied by a loss of liberty. The new
departure took three directions. It was definitely laid down that the
medical attendance afforded to the outdoor paupers was to be of
good quality, and thus necessarily above that obtained by the
poorest independent labourer, or even by the poor generally. This
was the outcome of a long campaign on behalf of the poorer
members of the medical profession, of which Wakley was the leader
in the House of Commons, and the Lancet the efficient organ.[375]
In
1853 the Poor Law Board considered that the qualifications of the
Poor Law medical officers ought to be such as to ensure for the
poor a degree of skill in their medical attendants equal to that which
can be commanded by the more fortunate classes of the
community.[376]
On the suggestion of the House of Commons
Committee on Poor Relief[377]
it was authoritatively enjoined on
boards of guardians in 1865 by a special circular that they were to
supply freely quinine, cod-liver oil, and other expensive medicines
to the sick poor;[378]
although it must have been plain that such
things were beyond the reach of the independent labourers
consulting the sixpenny doctor, and even beyond the usual
resources of the provident dispensaries of the period.[379]
Finally, in
1867, the Metropolitan Poor Act authorised the establishment
throughout London of Poor Law dispensaries. These institutions were](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-59-320.jpg)
![consistently pressed on the Metropolitan boards of guardians by the
Central Authority, as having been successful in Ireland in reducing
the amount of sickness among the poor, and as ensuring, not only
regular and more successful medical attention, but also a sufficient
supply of medicines and medical appliances of standard quality.[380]
By this elaborate systematisation of outdoor medical relief, the
Central Authority not only put within the reach of the sick paupers
medical attendance far superior to that accessible to the lowest
grade of independent labourers, but even placed the sick pauper in
the Metropolis, without loss of liberty, in a position equal to that of
the superior artisan subscribing to a good provident dispensary.
The most remarkable change of front was, however, that relating to
the institutional treatment of the sick. Down to 1847, it is not too
much to say that what may be called the hospital branch of Poor
Law administration[381]
was ignored alike by Parliament, public
opinion, and the Central Authority. We have shown that the
institutional provision for the sick was not so much as mentioned in
the Report of 1834, and that it remained practically ignored in all the
Orders, Circulars, and Reports of the Poor Law Commissioners. The
same is true of the first eighteen years of the Poor Law Board. Few
and far between are the incidental references to the sick wards of
the workhouses. There is not even a hint of a suggestion that relief
to the sick poor could most advantageously take the form of an offer
of the House. On the contrary, it was held in 1848 that applicants
for admission suffering from fever might even be refused
admission, the relieving officer being enjoined to find lodging
elsewhere for them,[382]
though how this was to be done the Central
Authority did not, in 1848, say. In 1857, the Metropolitan Boards of
Guardians were recommended to send such cases to the London
Fever Hospital[383]
(involving a payment by the guardians of 7s.
weekly). Finally, in 1864-5, we have an outburst of public
indignation, at the condition into which the sick wards of the
workhouses had been allowed to drift. The death of a pauper in
Holborn workhouse, and of another in St. Giles's workhouse, under
conditions which seemed to point to inhumanity and neglect, led to](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-60-320.jpg)
![an enquiry by three doctors (Anstie, Carr, and Ernest Hart),
commissioned by the Lancet newspaper, the formation of an
Association for improving the condition of the sick poor, and a
deputation to the Poor Law Board.[384]
The publication of various
reports on the workhouse infirmaries, in which terrible deficiencies
were revealed,[385]
led to public discussion and Parliamentary
debates. The Central Authority at once accepted the new standpoint.
It made no attempt to resist the provision of the necessarily costly
institutional treatment for the sick poor, whether or not their
ailments were infectious or otherwise dangerous to the public. The
progressive improvement of the hospital branch of Poor Law
administration, to use the phrase of the Central Authority itself,
which had in the preceding thirty years grown up unawares, was
now definitely accepted as an important feature of its policy.
Statutory powers were obtained for the provision of hospitals in the
Metropolis by combinations of boards of guardians. Urgent letters
were written pressing the boards of guardians to embark on the
expenditure required to enable them to provide efficiently for the
sick paupers.[386]
From 1865 onward, we see the Central Authority,
on the public-spirited initiative of Mr. W. Rathbone and the Liverpool
Select Vestry, pressing on the boards of guardians the employment
of salaried and qualified nurses to attend to the sick paupers,
whatever their complaints.[387]
We have even in 1867, so far as the
sick are concerned, the explicit disavowal by the Central Authority of
the very idea of the deterrent workhouse, which had formed so
prominent a part of the policy of 1834-1847. Mr. Gathorne Hardy,
speaking as President of the Poor Law Board, said there is one
thing ... which we must peremptorily insist on, namely, the
treatment of the sick in the infirmaries being conducted on an
entirely separate system, because the evils complained of have
mainly arisen from the workhouse management—which must to a
great degree be of a deterrent character—having been applied to the
sick, who are not proper objects for such a system.[388]
At first the new policy of the Central Authority for the institutional
treatment of the sick took the form of the erection of special](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-61-320.jpg)
![hospitals by Sick Asylum Districts.[389]
Presently, however, it came
to the conclusion that this involved an unnecessary expense, and
that it would be cheaper to revert to the idea of the Report of 1834,
and use the existing workhouse buildings by a system of
classification by institutions.[390]
So definitely was this recognised as
a reversion to 1834 that the Central Authority actually quoted the
passage of the 1834 Report in justification of its plan.[391]
From this
point may be dated the adoption of the policy of the provision, in
connection with the workhouse, but practically as a separate
institution, of what is now called the Poor Law Infirmary.[392]
In 1870
the Central Authority took pains to collect special statistics as to the
extent to which this recently developed provision for the sick was
being taken advantage of. It observes (and, significantly enough,
without expression of disapproval) that the numbers on the lists of
relieving officers may be swollen by poor persons who in previous
years, though really poor, refrained from coming on the rates, but
whom changes in the law or in the mode of its administration have
since attracted.[393]
Workhouses, it notes, originally designed
mainly as a test for the able-bodied, have, especially in the large
towns, been of necessity gradually transformed in to infirmaries for
the sick. The higher standard for hospital accommodation has had a
material effect upon the expenditure. So again it has been
considered necessary to attach to workhouses separate fever wards;
and wherever it was possible, these wards have been isolated by the
erection of a separate building.[394]
The extent to which the Poor
Law had become the public doctor was indeed remarkable. The
number of persons on outdoor relief who were actually sick, apart
from mere old age infirmity, and without their families, was found to
be 13 per cent of the whole, equal to about 119,000. The number in
the workhouses who were actually sick, irrespective of the vast
number of old people disabled by old age, but not actually upon the
sick list, varied in different unions from 14 to 39 per cent in the
provinces, and up to nearly 50 per cent in some Metropolitan
Unions; amounting, for the whole country, to about 60,000 actual
sick-bed cases.[395]
Taking indoor and outdoor patients together, the](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-62-320.jpg)
![total simultaneously under medical treatment in the twelfth week of
the half-year ending Lady Day 1870, was estimated at 173,000,
being three quarters of one per cent of the population, and perhaps
one out of four of all the persons under medical treatment in the
whole population. The story from this date is one continuous record,
on the one hand of an ever-increasing number of patients treated,
and, on the other, of never slackening pressure by the Central
Authority to induce apathetic or parsimonious boards of guardians to
expend money in making both the outdoor medical service and the
workhouse infirmaries as efficient and as well adapted and as well
equipped for the alleviation and cure of their patients—without the
least notion of the principle of less eligibility—as the most
scientifically efficient hospitals and State medical service in any part
of the world. After 1867, indeed, there was developed, for the
Metropolitan paupers suffering from infectious diseases, the splendid
hospital system of the Metropolitan Asylums Board.[396]
At the very
end of the existence of the Poor Law Board, Mr. Goschen seems
almost to have been contemplating a yet further extension. The
economical and social advantages, he observed, of free medicine
to the poorer classes generally as distinguished from actual paupers,
and perfect accessibility to medical advice at all times under
thorough organisation, may be considered as so important in
themselves as to render it necessary to weigh with the greatest care
all the reasons which may be adduced in their favour.[397]
F.—Persons of Unsound Mind
It is difficult to discover what was the policy of the Central Authority
during this period with regard to lunatics, idiots, and the mentally
defective. Lunacy had always been, and remained, a ground of
exception from the prohibition to grant outdoor relief. The provision
of a lodging for a lunatic was, moreover, an exception to the
prohibition of the payment of rent for a pauper. As a result of these
exceptions, there were on 1st January 1852, 4107 lunatics and idiots
on outdoor relief,[398]
and this number had increased by 1859 to
4892[399]
and by 1870 to 6199.[400]
The Central Authority took no](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-63-320.jpg)
![steps to require or persuade boards of guardians not to grant
outdoor relief to lunatics, nor yet to get any appropriate provision
made for them in the great general workhouses on which it had
insisted. Parliament in 1862 (in order to relieve the pressure on
lunatic asylums) expressly authorised arrangements to be made for
chronic lunatics to be permanently maintained in workhouses, under
elaborate provisions for their proper care.[401]
These arrangements
would have amounted, in fact, to the creation, within the
workhouse, of wards which were to be in every respect as well
equipped, as highly staffed, and as liberally supplied as a regular
lunatic asylum.[402]
The Central Authority transmitted the Act to the
boards of guardians, observing, with what almost seems like
sarcasm, that it was not aware of any workhouse in which any such
arrangements could conveniently be made;[403]
and the provisions of
this Act were, we believe, never acted upon. Whilst consistently
objecting to the retention in workhouses of lunatics who were
dangerous, or who were deemed curable, we do not find that the
Central Authority ever insisted on there being a proper lunatic ward
for the persons of unsound mind who were necessarily received, for
a longer or shorter period, in every workhouse.[404]
Moreover, the
Central Authority took no steps to get such persons removed to
lunatic asylums. In 1845 it had agreed with the Manchester Board of
Guardians (who did not want to make any more use of the county
asylum than they could help) that they were justified in retaining in
the workhouse any lunatics whom their own medical officer did not
consider proper to be confined in a lunatic asylum.[405]
In 1849 it
expressly laid it down that a weak-minded pauper or, as we now say,
a mentally defective, must either be a lunatic, and be certified and
treated as such, or not a lunatic, in which case no special treatment
could be provided for him or her in the one general workhouse to
which the Central Authority still adhered.[406]
We can find no
indication of policy as to whether it was recommended that such
mentally defectives should be granted outdoor relief, or (as one can
scarcely believe) required to inhabit a workhouse which made no
provision for them.[407]](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-64-320.jpg)
![The explanation of this paralysis of the Central Authority, as regards
the policy to be pursued with persons of unsound mind, is to be
found, we believe, in the existence and growth during this period of
the rival authority of the Lunacy Commissioners, who had authority
over all persons of unsound mind, whether paupers or not. The
Lunacy Commissioners had not habitually in their minds the principle
of less eligibility; and they were already, between 1848 and 1871,
making requirements with regard to the accommodation and
treatment of pauper lunatics that the Poor Law authorities regarded
as preposterously extravagant. The records of the boards of
guardians show visits of the inspectors of the Lunacy
Commissioners, and their perpetual complaints of the presence of
lunatics and idiots in the workhouses without proper
accommodation; mixed up with the sane inmates to the great
discomfort of both;[408]
living in rooms which the Lunacy
Commissioners considered too low and unventilated, with yards too
small and depressing, amid too much confusion and disorder, for the
section of the paupers for whom they were responsible.[409]
Such
reports, officially communicated to the Poor Law Board, seem to
have been merely forwarded for the consideration of the board of
guardians concerned. But other action was not altogether wanting.
Under pressure from the Lunacy Commissioners, the Central
Authority asked, in 1857, for more care in the conveyance of
lunatics;[410]
urged, in 1863, a more liberal dietary for lunatics in
workhouses;[411]
in 1867 it reminded the boards of guardians that
lunatics required much food, especially milk and meat;[412]
it was
thought very desirable that the insane inmates ... should have the
opportunity of taking exercise;[413]
it concurred with the Visiting
Commissioner in deeming it desirable that a competent paid nurse
should be appointed for the lunatic ward, in a certain workhouse;
[414]
it suggested the provision of leaning chairs in another
workhouse;[415]
and, in yet another, the desirability of not excluding
the persons of unsound mind from religious services.[416]
In 1870 it
issued a circular, transmitting the rules made by the Lunacy
Commissioners as to the method of bathing lunatics, for the careful](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-65-320.jpg)
![consideration of the boards of guardians.[417]
But we do not find that
the Central Authority issued any Order amending the General
Consolidated Order of 1847, which, it will be remembered, did not
include among its categories for classification either lunatics, idiots,
or the mentally defective; and the Central Authority did not require
any special provision to be made for them.
The policy of the Lunacy Commissioners was to get provision made
in every county for all the persons of unsound mind, whatever their
means, in specially organised lunatic asylums in which the best
possible arrangements should be made for their treatment and cure
irrespective of cost, and altogether regardless of making the
condition of the pauper lunatic less eligible than that of the poorest
independent labourer. Unlike the provision for education, and that
for infectious disease, the cost of this national (and as we may say
communistic) provision for lunatics was a charge upon the poor rate.
Under the older statutes, the expense of maintaining the inmates of
the county lunatic asylums was charged to the Poor Law authorities
of the parishes in which they were respectively settled; and the
boards of guardians were entitled to recover it, or part of it, from
any relatives liable to maintain such paupers, even in cases in which
the removal to the asylum was compulsory and insisted on in the
public interest.[418]
The great cost to the poor rate of lunatics sent to
the county lunatic asylums, and the difficulty of recovering the
amount from their relatives, prevented the whole-hearted adoption,
either by the boards of guardians, or the Central Authority, of the
policy of insisting on the removal of persons of unsound mind to the
county asylums. For the imbeciles and idiots of the Metropolitan
Unions, provision was made after 1867 in the asylums of the
Metropolitan Asylums Board.[419]
But no analogous provision for
those of other unions was made. The result was that, amid a great
increase of pauper lunacy, the proportion of the paupers of unsound
mind who were in lunatic asylums did not increase.[420]
On the other
hand the indisposition of the Central Authority to so amend the
General Consolidated Order of 1847 as to put lunatics in a separate
category, and require suitable accommodation and treatment for](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-66-320.jpg)
![them—an indisposition perhaps strengthened by the very high
requirements on which the Lunacy Commissioners would have
insisted—stood in the way of any candid recognition of the fact that
for thousands of lunatics, idiots, and mentally defectives, the
workhouse had, without suitable provision for them, and often to the
unspeakable discomfort of the other inmates, become a permanent
home.
G.—Defectives
During this period, the blind, the deaf and dumb, and the lame and
deformed were increasingly recognised by Parliament as classes for
whom the Poor Law authorities might, if they chose, provide
expensive treatment. This was done by authorising boards of
guardians, if they chose, to pay for their maintenance, whether
children or adults, in special institutions.[421]
We do not find that the
Central Authority suggested the adoption of this or any other policy
or gave any lead to the boards of guardians with regard to these
cases.[422]
H.—The Aged and Infirm
We have shown that neither the Report of 1834 nor the Central
Authority between 1834 and 1847 even suggested any departure
from the common practice of granting outdoor relief to the aged and
infirm. This continued, so far as the official documents show, to be
the policy of the Central Authority during the whole of the period
1847-1871.[423]
The only two references to the subject in the Orders
and Circulars of this period assume that the aged and infirm will
normally be relieved in their own homes. Thus, in 1852, in
commenting on the provision requiring the weekly payment of relief,
the Central Authority said, as to the cases in which the pauper is
too infirm to come every week for the relief, it is on many accounts
advantageous that the relieving officer should, as far as possible,
himself visit the pauper, and give the relief at least weekly.[424]
And
in the first edition of the Out-relief Regulation Order of 1852 (that of
25th August 1852) the Central Authority, far from prohibiting outdoor](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-67-320.jpg)
![relief to persons indigent and helpless from age, sickness, accident,
or bodily or mental infirmity, formally sanctioned this practice, by
ordering that one third at least of such relief should be given in
kind (viz., in articles of food or fuel, or in other articles of absolute
necessity),[425]
the object being expressly explained to be, not, as
might nowadays have been imagined, the discouragement of such
relief, but the prevention of its misappropriation.[426]
This provision
was objected to by boards of guardians up and down the country, on
the ground that it would be a hardship to the aged and infirm poor.
The Poplar Board of Guardians, for instance, stated that there are a
large number of persons under the denomination of aged and infirm
whom the guardians have, in their long practical experience, found it
expedient and not objectionable to relieve wholly in money, feeling
assured that it would be beneficially expended for their use, and that
in consequence of their infirmity the relieving officer or his assistant,
if necessary, is thereby enabled to conveniently relieve them at their
own house.[427]
The Norwich Guardians stated that it would be
difficult to determine (especially for the aged and sick poor) what
kind of food or articles should be given. They also communicated
with forty other unions, summoning them to concerted resistance.
[428]
A deputation from most of the large and populous unions in the
north of England ... and from several Metropolitan parishes,
representing in the aggregate upwards of 2,000,000 of population,
[429]
assembled in London, and objected to nearly all the provisions
of the Order.
Accompanied by about twenty-five members of Parliament, the
deputation waited on the Poor Law Board, and specially urged their
objection to being compelled to give a third of all outdoor relief in
kind. After two hours' argumentative discussion, Sir John Trollope
said that the board would reconsider the whole Order, which need
not in the meantime be acted upon; and he hinted at a probable
modification of the Article relating to relief in kind.[430]
In response to
these objections, the Central Authority does not seem even to have
suggested that outdoor relief to the aged and infirm was contrary to
its principles. It first intimated its willingness to modify the Order if](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-68-320.jpg)
![its working proved to be accompanied with hardship to the aged or
helpless poor[431]
and then within a few weeks withdrew the
provision altogether as regards any but the able-bodied.[432]
It was
expressly explained that the Order, as re-issued, was intended as a
precaution against the injurious consequences of maintaining out of
the poor rate able-bodied labourers and their families in a state of
idleness, and that the Central Authority left to the boards of
guardians full discretion as to the description of relief to be given to
indigent poor of every other class.[433]
From that date down to the
abolition of the Poor Law Board in 1871, we can find in the
documents no hint or suggestion that it disapproved of outdoor relief
to the aged and infirm. On 1st January 1871, nearly half the outdoor
relief was due to this cause.[434]
I.—Non-Residents
There was no change in the policy of preventing relief to paupers
not resident within the union. The Outdoor Relief Regulation Order
of 1852 embodied the prohibition with the same exceptions as had
been contained in the Outdoor Relief Prohibitory Order of 1844,
omitting, however, that of widows without children during the first
six months of their widowhood. But, as has been already mentioned,
at the very end of the period the Boarding-Out Orders of 1869, etc.,
permitted children to be maintained outside the union.
J.—The Workhouse
We have seen that between 1834 and 1847 the Central Authority
turned directly away from the express recommendations of the 1834
Report with regard to the institutional accommodation of the
paupers. Instead of a series of separate institutions appropriately
organised and equipped for the several classes of the pauper
population—the aged and infirm, the children, and the adult able-
bodied—the Central Authority had got established, in nearly every
union, one general workhouse; nearly everywhere the same cheap,
homely building, with one common regimen, under one
management, for all classes of paupers.](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-69-320.jpg)
![The justification for the policy which, as we have seen, Sir Francis
Head induced the Central Authority to substitute for the
recommendations of the 1834 Report, may have been his confident
expectation, in 1835, that the use of the workhouse was only to
serve as a test, which the applicants would not pass, and that
there was accordingly no need to regard the workhouse building as
a continuing home.[435]
This was the view taken by Harriet
Martineau, who, in her Poor Law Tales, describes the overseer of the
depauperised parish as locking the door of the empty workhouse
when it had completely fulfilled its purpose of a test by having made
all the applicants prefer and contrive to be independent of poor
relief. By 1847, however, it must have been clear that, even in the
most strictly administered parishes, under the most rigid application
of the Outdoor Relief Prohibitory Order, there would be permanently
residing in the workhouse a motley crowd of the aged and infirm
unable to live independently; the destitute chronic sick in like case;
the orphans and foundlings; such afflicted persons as the village
idiot, the senile imbecile, the deaf and dumb, and what we now call
the mentally defective; together with a perpetually floating
population of acutely sick persons of all ages; vagrants; girls with
illegitimate babies; wives whose husbands had deserted them, or
were in prison, in hospital, or in the Army or Navy; widows beyond
the first months of their widowhood and other women unable to
earn a livelihood; all sorts of ins and outs; and the children
dragging at the skirts of all these classes. The workhouse population
in 590 unions of England and Wales on 1st January 1849, was, in
fact, 121,331.[436]
The condition of these workhouse inmates, and
the character of the regimen to which they were subjected, had
been brought to public notice in 1847 in the notorious Andover case.
The insanitary condition of the workhouses of the period as places of
residence, and, in particular, their excessive death-rate, was
repeatedly brought to notice not only by irresponsible agitators, but
also by such competent statistical and medical critics as McCulloch
and Wakley.[437]
But the very idea of the general workhouse was now
subjected to severe criticism. During the last ten years, said the](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-70-320.jpg)
![author of an able book in 1852, I have visited many prisons and
lunatic asylums, not only in England, but in France and Germany. A
single English workhouse contains more that justly calls for
condemnation in the principle on which it is established than is
found in the very worst prisons or public lunatic asylums that I have
seen. The workhouse as now organised is a reproach and disgrace
peculiar to England; nothing corresponding to it is found throughout
the whole continent of Europe. In France the medical patients of our
workhouses would be found in 'hopitaux'; the infirm aged poor
would be in hospices; and the blind, the idiot, the lunatic, the
bastard child and the vagrant would similarly be placed each in an
appropriate but separate establishment. With us a common
Malebolge is provided for them all; and in some parts of the country
the confusion is worse confounded by the effect of Prohibitory
Orders, which, enforcing the application of the notable workhouse-
test, drive into the same common sink of so many kinds of vice and
misfortune the poor man whose only crime is his poverty, and whose
want of work alone makes him chargeable. Each of the buildings
which we so absurdly call a workhouse is, in truth (1) a general
hospital; (2) an almshouse; (3) a foundling house; (4) a lying-in
hospital; (5) a school house; (6) a lunatic asylum; (7) an idiot
house; (8) a blind asylum; (9) a deaf and dumb asylum; (10) a
workhouse; but this part of the establishment is generally a lucus a
non lucendo, omitting to find work even for able-bodied paupers.
Such and so varied are the destinations of these common
receptacles of sin and misfortune, of sorrow and suffering of the
most different kinds, each tending to aggravate the others with
which it is unnecessarily and injuriously brought into contact. It is at
once equally shocking to every principle of reason and every feeling
of humanity, that all these varied forms of wretchedness should be
thus crowded together into one common abode, that no attempt
should be made by law to classify them, and to provide appropriate
places for the relief of each.[438]
During the period now under review, 1847-71, we see the Central
Authority becoming gradually alive to the draw-backs of this mixture](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-71-320.jpg)
![of classes. At first its remedy seems to have been to take particular
classes out of the workhouse. We have already described the
constant attempts, made from the very establishment of the Poor
Law Board, to have the children removed to separate institutions and
to get the vagrants segregated into distinct casual wards. It was the
resistance and apathy of the boards of guardians that prevented
these attempts being particularly successful,[439]
and the Central
Authority appears not to have felt able to issue peremptory orders
on the subject. The policy of the Lunacy Commissioners drew many
lunatics out of the workhouses, but this was more than made up by
the increasing tendency to seclude the village idiot, so that the
workhouse population of unsound mind actually increased.
We do not find that there was during the whole period any alteration
of the General Consolidated Order of 1847, upon which the regimen
of the workhouse depended. In spite of the increasing number of
the sick and the persons of unsound mind, the seven classes of
workhouse inmates determined by that Order were adhered to, and
received no addition, though the Poor Law Board favoured the sub-
division of these classes so far as it was reasonably possible in the
existing buildings, especially in the case of women. In a letter of
1854[440]
it lamented the evil which arose from the association of
girls, when removed from workhouse union schools, with women of
bad character in the able-bodied women's ward, and wished that it
could be prevented. At the same time it stated that in the smaller
workhouses it was often impracticable to provide the
accommodation which would be necessary in order to maintain a
complete separation; and while pointing out that it was legally
competent for the guardians (with its approval) to erect extra
accommodation, by means of which this contamination could be
avoided, the Central Authority did not even remotely suggest that it
was the guardians' duty so to do. By 1860 it had given instructions
that every new workhouse should be so constructed as to allow of
the requisite classification.[441]](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-72-320.jpg)
![From about 1865 onwards we note a new spirit in all the circulars
and letters relating to the workhouse. The public scandal caused by
the Lancet inquiry into the conditions of the sick poor in the
workhouses, and the official reports and Parliamentary discussions
that ensued, seem to have enabled the Central Authority to take up
a new attitude with regard both to workhouse construction and
workhouse regimen. From this time forth the workhouse is
recognised as being, not merely a test of destitution for the able-
bodied, which they were not expected long to endure, but also the
continuing home of large classes of helpless and not otherwise than
innocent persons. Able-bodied people, reported the Medical Officer
in 1867, are now scarcely at all found in them during the greater
part of the year.... Those who enjoy the advantages of these
institutions are almost solely such as may fittingly receive them, viz.
the aged and infirm, the destitute sick and children. Workhouses are
now asylums and infirmaries.[442]
From now onwards we see the Central Authority always striving to
improve the workhouse. In the Circulars of 1868 much attention was
paid to the sufficiency of space and ventilation. It was required that
parallel blocks of building should be so far apart as to allow free
access to light and air; blocks connected at a right or acute angle
were to be avoided.
Ordinary wards were to be at least ten feet high and eighteen feet
wide, the length depending on the number of inmates; 300 cubic
feet of space were required for each healthy person in a dormitory,
500 for infirm persons able to leave the dormitory during the day,
and 700 in a day and night room.[443]
The Visiting Committee was to
ascertain not merely whether the total number for which the
workhouse is certified has been exceeded, but whether the number
of any one class exceeds the accommodation available for it.[444]
No
wards were to be placed side by side without a corridor between
them; the corridors were to be six feet wide, and ordinary
dormitories were to have windows into them. Windows and fanlights
into internal spaces were to be made to open to be used as](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-73-320.jpg)
![ventilators, and ventilation was also to be effected by special
means, apart from the usual means of doors, windows, and fire-
places, air-bricks being recommended as a simple method.[445]
No
rooms occupied by the inmates as sleeping-rooms were to be on the
boundary of the workhouse site. Hot and cold water was to be
distributed to the bath-rooms and sick wards. Airing yards for the
inmates were to be of sufficient size—with a rider that if partially
or wholly paved with stone or brick or asphalted or gas-tarred they
are often better than if covered with gravel.[446]
Yards for the
children, sick, and aged were to be enclosed with dwarf walls and
palisades where practicable, presumably with the object of giving a
look-out, and making the yard slightly less prison-like.[447]
Small
yards, and a work-room, and a covered shed for working in bad
weather, were to be provided for vagrants.[448]
For workhouses
having a large number of children the Poor Law Board
recommended, in addition to the school-rooms, day-rooms, covered
play-sheds in their yards, and industrial work-rooms.[449]
The
staircases were to be of stone; the timber, Baltic fir and English oak;
fire escapes were to be provided; these and many other details were
laid down, all tending to make the building solid and capacious.[450]
There was no mention of ornament, no regard to appearance, no
hint that anything might be done to relieve the dead ugliness of the
place; but it must be recognised that the Central Authority had, by
1868, travelled far from the low, cheap, homely building which it
was recommending thirty years before.[451]
Separate dormitories, day-rooms, and yards (apparently not dining-
rooms) were required for the aged, able-bodied, children, and sick of
each sex, and these were the only divisions laid down as
fundamental, but the Circular went on to recommend provision (1)
so far as practicable for the sub-division of the able-bodied women
into two or three classes with reference to moral character, or
behaviour, the previous habits of the inmates, or such other grounds
as might seem expedient, and (2) in the larger workhouses for
the separate accommodation of the following classes of sick—](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-74-320.jpg)
![Ordinary sick of both sexes.
Lying-in women, with separate labour room.
Itch cases of both sexes.
Dirty and offensive cases of both sexes.
Venereal cases of both sexes.
Fever and smallpox cases of both sexes (to be in a separate
building with detached rooms).
Children (in whose case sex was not mentioned).[452]
In the furnishing of the wards the simplicity of 1868 was equally far
removed from that of 1835. Ordinary dormitories contained beds 2
feet 6 inches wide, chairs, bells, and gas where practicable. Day-
rooms were to have an open fireplace, benches, cupboards (or open
shelves, which were preferred), tables, gas, combs, and hairbrushes.
A proportion of chairs were to be provided for the aged and
infirm; and of the benches, likewise, those for the aged and infirm
should have backs, and be of sufficient width for reasonable
comfort. In the dining-rooms were to be benches, tables, a
minimum of necessary table utensils, and if possible gas and an
open fireplace. The sick wards were to be furnished with more care,
and with an eye to medical efficiency. It is unnecessary to go into
the long and detailed list of the medical appliances which were
required. There is even some notice of appearances in a suggestion
that cheerful-looking rugs should be placed on the beds, and of
comfort in the arm and other chairs for two-thirds of the number of
the sick. There were also to be short benches with backs, and (but
these only for special cases) even cushions; rocking-chairs for the
lying-in wards, and little arm-chairs and rocking-chairs for the
children's sick wards.[453]
Dr. Smith had further recommended a Bible
for each inmate, entertaining illustrated and religious periodicals,
tracts and books, games, and a foot valance to the bed to add to
the appearance of comfort,[454]
These suggestions were not
specifically taken up by the Central Authority, but Dr. Smith's report
was circulated to the guardians, without comment.[455]
We have the
beginning, too, between 1863 and 1867, of the improvement of the
food, which was regulated in each workhouse by a separate Special](https://image.slidesharecdn.com/1022999-250522042315-d456be88/85/Recent-Perspectives-In-Random-Matrix-Theory-And-Number-Theory-Mezzadri-F-75-320.jpg)
Recent Perspectives In Random Matrix Theory And Number Theory Mezzadri F
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- 5. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom The title below are available from booksellers, or from Cambridge University Press at www.cambridge.org 161 Lectures on block theory, BURKHARD KÜLSHAMMER 163 Topics in varieties of group representations, S.M. VOVSI 164 Quasi-symmetric designs, M.S. SHRlKANDE & S.S. SANE 166 Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) 168 Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) 169 Boolean function complexity, M.S. PATERSON (ed) 170 Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNK 171 Squares, A.R. RAJWADE 172 Algebraic varieties, GEORGE R. KEMPF 173 Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) 174 Lectures on mechanics, J.E. MARSDEN 175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) 176 Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) 177 Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) 178 Lower K-and L-theory, A. RANlCKl 179 Complex projective geometry, G. ELLlNGSRUD et al 180 Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT 181 Geometric group theory I, G.A. NlBLO & M.A. ROLLER (eds) 182 Geometric Group Theory II, G.A. NlBLO & M.A. ROLLER (eds) 183 Shintani Zeta Functions, A. YUKlE 184 Arithmetical Functions, W. SCHWARZ & J. SPlLKER 185 Representations of solvable groups. O. MANZ & T.R. WOLF 186 Complexity: knots, colotigs and counting, D.J.A. WELSH 187 Surveys in combinator. 1993 K. WALKER (ed) 188 Local’analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN 189 Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY 190 Polynomial inwariants of finite groups, D.J. BENSON 191 Finite geometry and combinatorics, F. DE CLERCK et al 192 Symplectic geometry, D. SALAMON (ed) 194 lndependent random variables and rearrangment invariant spaces, M. BRAVERMAN 195 Arithmetic of blowup algebras, WOLMER VASCONCELOS 196 Microlocal analysis for differential operators, A. GRlGlS & J. SJÖSTRAND 197 Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI et al 198 The algebraic characterization of geometric 4-manifolds, J.A. HlLLMAN 199 Invariant Potential theory in rhe unit ball of Cn , MANFRED STOLL 200 The Grothendieck theory of dessins d’enfant, L. SCHNEPS (ed) 201 Singularities, JEAN-PAUL BRASSELET (ed) 202 The technique of pseudodifferential operators, H.O. CORDES 203 Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMlTH 204 Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWEB (eds) 205 Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) 207 Groups of Lie type and their geometries, W.M. KANTOR & L. Dl MARTINO (eds) 208 Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) 209 Arithmetic of diagonal hypersutrfaces over finite fields, Q. GOUVÉA & N. U 210 Hilbert C*-modules, E.C. LANCE 211 Groups 93 Galway / St Andrews I, CM. CAMPBELL et al (eds) 212 Groups 93 Galway / St Andrews II, C.M. CAMPBELL et al (eds) 214 Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO el al 215 Number theory 1992–93, S. DAVID (ed) 216 Stochastic partial differential equations, A. ETHERlDGE (ed) 217 Quadratic form wirh applications to algebraic geometry and topology, A. PFlSTER 218 Surveys in-combinatorics, 1995, PETER ROWLINSON (cd) 220 Algebraic set theory, A. JOYAL & I. MOERDIJK 221 Harmonic approximation, S.J. GARDINER 222 Advances in linear logic, J.-Y. GlRARD. Y. LAFONT & L. REGNIER (eds) 223 Analytic semigroups and semilinear initial boundary value problems, KAZUAKl TAIRA 224 Comrutabilitv, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAlNER (eds) 225 A mathematical introduction to string theory, S. ALBEVERIO et al 226 Novikov conjectures, index theorems and rigidity I. S. FERRY, A. RANlCKI & 1. ROSENBERG (eds) 227 Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANlCK1 & J. ROSENBERG (eds) 228 Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds) 229 Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK 230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN 231 Semigmup theory and its applications. K.H. HOFMANN & M.W. MISLOVE (eds) 232 The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS 233 Finite fields and applications, S. COHEN & H. NIDERREITER (eds) 234 Introduction to subfactors, V. JONES & V.S. SUNDER 235 Number theory 1993–94. S. DAVID (ed) 236 The James forest, H. FETTER & B. GAMBOA DE BUEN
- 6. 237 Sieve methods. exponential sums, and their applications in number theory, G.R.H. GREAVES et al 238 Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) 240 Stable groups, FRANK 0. WAGNER 241 Surveys in combinatorics, 1997. R.A. BAILEY (ed) 242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) 243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) 244 Model theory of groups and automrphism groups, D. EVANS (ed) 245 Geometry, combinatorial designs and related structures, J.W.P HIRSCHFELD et al 246 p-Automorphisms of Finite-groups, E. KHUKHRO 247 Analytic number theory, Y. MOTOHASHI (ed) 248 Tame topology and o-minimal structures, LOU VAN DEN DRIES 249 The atlas of finite gmups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds) 250 Characters and blocks of finite groups. G. NAVARRO 251 Groner bases and applications, B. BUCHBERGER & E WINKLER (eds) 252 Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO, R. STÖHR (eds) 253 The q-Schur algebra, S. DONKIN 254 Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) 255 Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) 256 Aspects of Galois theory, HELMUT VOLKLBIN et al 257 An introduction to nancommutative differential geometty and its physical applications 2ed, J. MADORE 258 Sets and proofs, S.B. COOPER & J. TRUSS (eds) 259 Models and computability, S.B. COOPER & J. TRUSS (eds) 260 Groups St Andrews 1997 in Bath, I, CM. CAMPBELL et al 261 Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al 262 Analysis and logic, C.W. HENSON, J. IOVINO, A.S. KECHRIS & E. ODELL 263 Singularity theoy, BILL BRUCE & DAVID MOND (eds) 264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) 265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART 267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds) 268 Spectral asymptatics in the semi-classical limit, M. DIMASSI & J. SJÖSTRAND 269 Ergodic theory and topological dynamics, M.B. BEKKA & M. MAYER 270 Analysis on Lie groups, N.T. VAROPOULOS & S. MUSTAPHA 271 Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV 272 Character theory for the odd order theorem, T. PETERFALVI 273 Spectral theory geometry, E.B. DAVIES & Y. SAFAROV (eds) 274 The Mandlebrot set, theme and variations, TAN LEI (ed) 275 Descriptive set theory and dynamical systems, M. FOREMAN et al 276 Singularities of plane curves, E. CASAS-ALVERO 277 Computatianal and geometric aspects of modern algebra, M.D. ATKINSON et al 278 Glbal attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO 279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds) 280 Chamctw and automorphism groups of compact Riemann surfaces, THOMAS BREUER 281 Explicit birational geometry of 3-folds, ALESSIO CORTI & MlLES REID (eds) 282 Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO 283 Nonlinear elasticity, Y. FU & R.W. OGDEN (eds) 284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SÜLI (eds) 285 Rational points on curves over finite fields. H. NIEDERREITER & C. XING 286 Clifford algebras and spinors 2ed, p. LOUNESTO 287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA & E. MARTINEZ (eds) 288 Surveys in cambinatorics, 2001, J. HIRSCHFELD (ed) 289 Aspects of Sobolev-type inequalities. L. SALOFF-COSTE 290 Quantum groups and Lie theory, A. PRESSLEY (ed) 291 Tits buildings and the model theory of groups, K. TENT (ed) 292 A quantum groups primer, S. MAJID. 293 Second order partial diffetcntial equations in Hilbett spaces, G. DA PRATO & I. ZABCZYK 294 Imwduction to the theory of operator spaces, G. PISIER 295 Geometry and integrability. LIONEL MASON & YAVUZ NUTKU (eds) 296 Lectures on invariant theory, IGOR DOLGACHEV 297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC, & C. SERIES (eds) 300 Introduction to Möbius differential geometry, UDO HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A. JOHNSON 302 Diicrete and continuous nonlinear Schrödinger systems, M.J. ABLOWITZ, B. PRINARI, & A.D. TRUBATCH 303 Number theory and algebraic geometry, MILES REID & ALEXEI SKOROBOOATOV (eds) 304 Groups St Andrews 2001 in Oxford Vol. I, COLIN CAMPBELL, EDMUND ROBERTSON 305 Groups St Andrews 2001 in Oxford Vol. 2. C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 307 Surveys in combinatorics 2003, C.D. WENSLEY (ed) 308 Topology, Geometry and Quantum Field Theory, U. TILLMAN (ed) 309 Corings and comodules, TOMASZ BRZEZINSKI & ROBERT WISBAUER 310 Topics in dynamics and ergodic theory, SERGEY BEZUGLYI & SERGIY KOLYADA (eds) 311 Groups, T.W. MÜLLER 312 Foundations of computational mathematics, Minneapolis 2002, FELIPE CUCKER et al (eds) 313 Transcendental Aspects of Algebraic Cycles, S. MÜLLER-STACH & C. PETERS (eds)
- 7. London Mathematical Society Lecture Note Series. 322 Recent Perspectives in Random Matrix Theory and Number Theory Edited by F. MEZZADRI N. C. SNAITH University of Bristol
- 8. CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521620581 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2005 A catalogue record for this publication is available from the British Library ISBN 978-0-521-62058-1 paperback Transferred to digital printing (with corrections) 2009 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.
- 9. v Contents Introduction vii F. Mezzadri and N.C. Snaith Prime number theory and the Riemann zeta-function 1 D. R. Heath-Brown Introduction to the random matrix theory: Gaussian Uni- tary Ensemble and beyond 31 Yan V. Fyodorov Notes on pair correlation of zeros and prime numbers 79 D.A. Goldston Notes on eigenvalue distributions for the classical compact groups 111 Brian Conrey Compound nucleus resonances, random matrices, quantum chaos 147 Oriol Bohigas Basic analytic number theory 185 David W. Farmer Applications of mean value theorems to the theory of the Riemann zeta function 201 S.M. Gonek Families of L-functions and 1-level densities 225 Brian Conrey L-functions and the characteristic polynomials of random matrices 251 J.P. Keating Spacing distributions in random matrix ensembles 279 Peter J. Forrester Toeplitz determinants, Fisher-Hartwig symbols, and ran- dom matrices 309 Estelle L. Basor Mock-Gaussian behaviour 337 C.P. Hughes
- 10. vi Some specimens of L-functions 357 Philippe Michel Computational methods and experiments in analytic num- ber theory 425 Michael Rubinstein
- 11. Introduction F. Mezzadri and N. C. Snaith This volume of proceedings stems from a school that was part of the programme Random Matrix Approaches in Number Theory, which ran at the Isaac Newton Institute for Mathematical Sciences, Cambridge, from 26 January until 16 July 2004. The purpose of these proceedings is twofold. Firstly, the impressive recent progress in analytic number theory brought about by the introduction of random matrix techniques has created a rapidly developing area of research. As a consequence there is not as yet a textbook on the subject. This volume is intended to fill this gap. There are, of course, well-established texts in both random matrix theory and analytic number theory, but very few of them treat in any length or detail these new applications of random matrix theory. Secondly, this new branch of mathematics is intrinsically multidisciplinary; teaching young researchers in random matrix theory, mathematical physics and number theory mathematical techniques that are not a natural part of their education is essential to introduce a new generation of scientists to this important and rapidly developing field. In writing their contributions to the proceedings, the lecturers kept in mind the diverse backgrounds of the audience to whom this volume is addressed. The material in the volume includes the basic techniques of random matrix theory and number theory needed to understand the most important achieve- ments in the subject; it also gives a comprehensive survey of recent results where random matrix theory has played a major role in advancing our under- standing of open problems in number theory. We hope that the choice of topics will be useful to both the advanced graduate student and to the established researcher. These proceedings contain a set of introductory lectures to analytic number theory and random matrix theory, written by Roger Heath-Brown and Yan Fy- odorov respectively. The former includes a survey of elementary prime number theory and an introduction to the theory of the Riemann zeta function and other L-functions, while Fyodorov’s lectures provide the reader with one of the main tools used in the theory of random matrices: the theory of orthogonal polynomials. This ubiquitous technique is then applied to the computation of the spectral correlation functions of eigenvalues of the Hermitian matrices which form the Gaussian Unitary Ensemble (GUE), as well as to comput- ing the averages of moments and ratios of characteristic polynomials of these
- 12. viii F. Mezzadri and N.C. Snaith Hermitian matrices. In contrast, fundamental techniques for calculating vari- ous eigenvalue statistics on ensembles of unitary matrices can be found in the “Notes on eigenvalue distributions for the classical compact groups” by Brian Conrey. These are the groups of matrices that are used in connection with L- functions, for example in the lectures of Hughes and Keating. The articles of Peter Forrester and Estelle Basor discuss more specific topics in random matrix theory. Forrester reviews in detail the theory of spacing distributions for var- ious ensembles of matrices and emphasizes its connections with the theory of Painlevé equations and of Fredholm determinants; Basor’s lectures introduce the reader to the theory of Toeplitz determinants, their asymptotic evaluations for both smooth and singular symbols and their connection to random matrix theory. Dan Goldston reviews how random matrix theory and number theory came together unexpectedly when Montgomery, assuming the Riemann hypothesis, conjectured the two-point correlation function of the Riemann zeros, which Dyson recognized it as the two-point correlation function for eigenvalues of the random matrices in the CUE (or, equivalently, the GUE) ensemble. Looking toward applications to physics, Oriol Bohigas’s article gives an historical survey of how random matrix theory was instrumental in the understanding of the statistical properties of spectra of complex nuclei and of individual quantum mechanical systems whose classical limit exhibits chaotic behaviour. After Montgomory’s discovery overwhelming numerical evidence, largely produced by Andrew Odlyzko in the late 1980s, supported the hypothesis that the non- trivial zeros of the Riemann zeta function are locally correlated like eigenvalues of random matrices in the GUE ensemble. Later Hejhal (1994), and then Rudnick and Sarnak (1994,1996) proved similar results for the three and higher point correlations. Several lectures are devoted to specific and more advanced topics in number theory. David Farmer introduces the reader to techniques in analytic number theory, discussing various ways to manipulate Dirichlet series, while Steve Gonek extends this to discuss mean-value theorems and their applications. Philippe Michel discusses the construction of many examples of L-functions, including those associated to elliptic curves and modular forms. The remaining lectures highlight the connection between L-functions and random matrix theory. Brian Conrey’s lectures “Families of L-functions and 1-level densities” concern the statistics of zeros of families of L-functions near the point where the line on which their Riemann hypothesis places their zeros crosses the real axis. Based on the example of the function field zeta functions, these statistics were proposed by Katz and Sarnak (1999) to be those of the eigenvalues of one of the classical compact groups, namely U(N), USp(2N) and O(N). The lectures of Jon Keating reveal how the local statistical properties of the Riemann zeta function and other L-functions are inherently determined by the distribution of their zeros, thus high up the critical line ζ(s) can be modelled by the characteristic polynomial of random matrices belonging to
- 13. Introduction ix U(N). As a consequence of this property, techniques well developed in ran- dom matrix theory can lead to conjectures for quantities like moments and distributions of the values of L-functions, which have been open problems for almost eighty years. Chris Hughes discusses how the first few moments of the smooth counting functions of the eigenvalues of random matrices and of the zeros of L-functions are Gaussian while their distributions are not. Since much of the predictive power of random matrix theory is based on conjectures, numerical experiments play an important role in the theory; Michael Rubin- stein’s article introduces the reader to the most important techniques used in computational number theory and to conjectures and numerical experiments connecting number theory with random matrix theory. We are particularly grateful to David Farmer and Brian Conrey for care- fully reading many of the articles and to the staff of the Newton Institute for their invaluable assistance in making the school such a successful event. We also thankfully acknowledge financial contributions from the EU Network ‘Mathematical Aspects of Quantum Chaos’, the Institute of Physics Publish- ing, the Isaac Newton Institute for the Mathematical Sciences and the US National Science Foundation. Francesco Mezzadri and Nina C. Snaith July 2004 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
- 15. Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown 1 Primes An integer p ∈ N is said to be “prime” if p = 1 and there is no integer n dividing p with 1 n p. (This is not the algebraist’s definition, but in our situation the two definitions are equivalent.) The primes are multiplicative building blocks for N, as the following crucial result describes. Theorem 1. (The Fundamental Theorem of Arithmetic.) Every n ∈ N can be written in exactly one way in the form n = pe1 1 pe2 2 . . . pek k , with k ≥ 0, e1, . . . , ek ≥ 1 and primes p1 p2 . . . pk. For a proof, see Hardy and Wright [5, Theorem 2], for example. The situation for N contrasts with that for arithmetic in the set {m + n √ −5 : m, n ∈ Z}, where one has, for example, 6 = 2 × 3 = (1 + √ −5) × (1 − √ −5),
- 16. 2 D. R. Heath-Brown with 2, 3, 1 + √ −5 and 1 − √ −5 all being “primes”. A second fundamental result appears in the works of Euclid. Theorem 2. There are infinitely many primes. This is proved by contradiction. Assume there are only finitely many primes, p1, p2, . . . , pn, say. Consider the integer N = 1 + p1p2 . . . pn. Then N ≥ 2, so that N must have at least one prime factor p, say. But our list of primes was supposedly complete, so that p must be one of the primes pi, say. Then pi divides N − 1, by construction, while p = pi divides N by assump- tion. It follows that p divides N − (N − 1) = 1, which is impossible. This contradiction shows that there can be no finite list containing all the primes. There have been many tables of primes produced over the years. They show that the detailed distribution is quite erratic, but if we define π(x) = #{p ≤ x : p prime}, then we find that π(x) grows fairly steadily. Gauss conjectured that π(x) ∼ Li(x), where Li(x) = x 2 dt log t , that is to say that lim x→∞ π(x) Li(x) = 1. The following figures bear this out. π(108 ) = 5,776,455 π(x) Li(x) = 0.999869147. . . , π(1012 ) = 37,607,912,018 π(x) Li(x) = 0.999989825. . . , π(1016 ) = 279,238,341,033,925 π(x) Li(x) = 0.999999989. . . . It is not hard to show that in fact Li(x) ∼ x log x , but it turns out that Li(x) gives a better approximation to π(x) than x/ log x does. Gauss’ conjecture was finally proved in 1896, by Hadamard and de la Vallée Poussin, working independently. Theorem 3. (The Prime Number Theorem.) We have π(x) ∼ x log x as x → ∞.
- 17. Prime number theory and the Riemann zeta-function 3 One interesting interpretation of the Prime Number Theorem is that for a number n in the vicinity of x the “probability” that n is prime is asymptotically 1/ log x, or equivalently, that the “probability” that n is prime is asymptot- ically 1/ log n. Of course the event “n is prime” is deterministic — that is to say, the probability is 1 if n is prime, and 0 otherwise. None the less the probabilistic interpretation leads to a number of plausible heuristic arguments. As an example of this, consider, for a given large integer n, the probability that n + 1, n + 2, . . . , n + k are all composite. If k is at most n, say, then the probability that any one of these is composite is about 1−1/ log n. Thus if the events were all independent, which they are not, the overall probability would be about 1 − 1 log n k . Taking k = µ(log n)2 and approximating 1 − 1 log n log n by e−1 , we would have that the probability that n + 1, n + 2, . . . , n + k are all composite, is around n−µ . If En is the event that n + 1, n + 2, . . . , n + k are all composite, then the events En and En+1 are clearly not independent. However we may hope that En and En+k are independent. If the events En were genuinely independent for different values of n then an application of the Borel-Cantelli lemma would tell us that En should happen infinitely often when µ 1, and finitely often for µ ≥ 1. With more care one can make this plausible even though En and En are correlated for nearby values n and n . We are thus led to the following conjecture. Conjecture 1. If p denotes the next prime after p then lim sup p→∞ p − p (log p)2 = 1. Numerical evidence for this is hard to produce, but what there is seems to be consistent with the conjecture. In the reverse direction, our simple probabilistic interpretation of the Prime Number Theorem might suggest that the probability of having both n and n+1 prime should be around (log n)−2 . This is clearly wrong, since one of n and n+1 is always even. However, a due allowance for such arithmetic effects leads one to the following. Conjecture 2. If c = 2 p2 1 − 1 (p − 1)2 = 1.3202 . . . ,
- 18. 4 D. R. Heath-Brown the product being over primes, then #{n ≤ x : n, n + 2 both prime} c x 2 dt (log t)2 . (1.1) The numerical evidence for this is extremely convincing. Thus the straightforward probabilistic interpretation of the Prime Number Theorem leads to a number of conjectures, which fit very well with the available numerical evidence. This probabilistic model is known as “Cramér’s Model” and has been widely used for predicting the behaviour of primes. One further example of this line of reasoning shows us however that the primes are more subtle than one might think. Consider the size of π(N + H) − π(N) = #{p : N p ≤ N + H}, when H is small compared with N. The Prime Number Theorem leads one to expect that π(N + H) − π(N) N+H N dt log t ∼ H log N . However the Prime Number Theorem only says that π(x) = x 2 dt log t + o( x log x ), or equivalently that π(x) = x 2 dt log t + f(x), where f(x) x/ log x → 0 as x → ∞. Hence π(N + H) − π(N) = N+H N dt log t + f(N + H) − f(N). In order to assert that f(N + H) − f(N) H/ log N → 0 as N → ∞ we need cN ≤ H ≤ N for some constant c 0. None the less, considerably more subtle arguments show that π(N + H) − π(N) ∼ H log N even when H is distinctly smaller than N. A careful application of the Cramér Model suggests the following conjec- ture.
- 19. Prime number theory and the Riemann zeta-function 5 Conjecture 3. Let κ 2 be any constant. Then if H = (log N)κ we should have π(N + H) − π(N) ∼ H log N as N → ∞. This is supported by the following result due to Selberg in 1943 [15]. Theorem 4. Let f(N) be any increasing function for which f(N) → ∞ as N → ∞. Assume the Riemann Hypothesis. Then there is a subset E of the integers N, with #{n ∈ E : n ≤ N} = o(N) as N → ∞, such that π(n + f(n) log2 n) − π(n) ∼ f(n) log n for all n ∈ E. Conjecture 3 would say that one can take E = ∅ if f(N) is a positive power of log N. Since Cramér’s Model leads inexorably to Conjecture 3, it came as quite a shock to prime number theorists when the conjecture was disproved by Maier [9] in 1985. Maier established the following result. Theorem 5. For any κ 1 there is a constant δκ 0 such that lim sup N→∞ π(N + (log N)κ ) − π(N) (log N)κ−1 ≥ 1 + δκ and lim inf N→∞ π(N + (log N)κ ) − π(N) (log N)κ−1 ≤ 1 − δκ. The values of N produced by Maier, where π(N + (log N)κ ) − π(N) is abnormally large, (or abnormally small), are very rare. None the less their existence shows that the Cramér Model breaks down. Broadly speaking one could summarize the reason for this failure by saying that arithmetic effects play a bigger rôle than previously supposed. As yet we have no good alternative to the Cramér model. 2 Open Questions About Primes, and Important Results Here are a few of the well-known unsolved problems about the primes. (1) Are there infinitely many “prime twins” n, n+2 both of which are prime? (Conjecture 2 gives a prediction for the rate at which the number of such pairs grows.)
- 20. 6 D. R. Heath-Brown (2) Is every even integer n ≥ 4 the sum of two primes? (Goldbach’s Conjec- ture.) (3) Are there infinitely many primes of the form p = n2 + 1? (4) Are there infinitely many “Mersenne primes” of the form p = 2n − 1? (5) Are there arbitrarily long arithmetic progressions, all of whose terms are prime? (6) Is there always a prime between any two successive squares? However there have been some significant results proved too. Here are a selection. (1) There are infinitely many primes of the form a2 + b4 . (Friedlander and Iwaniec [4], 1998.) (2) There are infinitely many primes p for which p + 2 is either prime or a product of two primes. (Chen [2], 1966.) (3) There is a number n0 such that any even number n ≥ n0 can be written as n = p+p with p prime and p either prime or a product of two primes. (Chen [2], 1966.) (4) There are infinitely many integers n such that n2 + 1 is either prime or a product of two primes. (Iwaniec [8], 1978.) (5) For any constant c 243 205 = 1.185 . . ., there are infinitely many integers n such that [nc ] is prime. Here [x] denotes the integral part of x, that is to say the largest integer N satisfying N ≤ x. (Rivat and Wu [14], 2001, after Piatetski-Shapiro, [11], 1953.) (6) Apart from a finite number of exceptions, there is always a prime between any two consecutive cubes. (Ingham [6], 1937.) (7) There is a number n0 such that for every n ≥ n0 there is at least one prime in the interval [n , n + n0.525 ]. (Baker, Harman and Pintz, [1], 2001.) (8) There are infinitely many consecutive primes p, p such that p − p ≤ (log p)/4. (Maier [10], 1988.) (9) There is a positive constant c such that there are infinitely many consec- utive primes p, p such that p − p ≥ c log p (log log p)(log log log log p) (log log log p)2 . (Rankin [13], 1938.)
- 21. Prime number theory and the Riemann zeta-function 7 (10) For any positive integer q and any integer a in the range 0 ≤ a q, which is coprime to q, there are arbitrarily long strings of consecutive primes, all of which leave remainder a on division by q. (Shiu [16], 2000.) By way of explanation we should say the following. The result (1) demon- strates that even though we cannot yet handle primes of the form n2 + 1, we can say something about the relatively sparse polynomial sequence a2 + b4 . The result in (5) can be viewed in the same context. One can think of [nc ] as being a “polynomial of degree c” with c 1. Numbers (2), (3) and (4) are approximations to, respectively, the prime twins problem, Goldbach’s prob- lem, and the problem of primes of the shape n2 + 1. The theorems in (6) and (7) are approximations to the conjecture that there should be a prime between consecutive squares. Of these (7) is stronger, if less elegant. Maier’s result (8) shows that the difference between consecutive primes is sometimes smaller than average by a factor 1/4, the average spacing being log p by the Prime Number Theorem. (Of course the twin prime conjecture would be a much stronger result, with differences between consecutive primes sometimes being as small as 2.) Similarly, Rankin’s result (9) demonstrates that the gaps between consecutive primes can sometimes be larger than average, by a factor which is almost log log p. Again this is some way from what we expect, since Conjecture 1 predict gaps as large as (log p)2 . Finally, Shiu’s result (10) is best understood by taking q = 107 and a = 7, 777, 777, say. Thus a prime leaves remainder a when divided by q, precisely when its decimal expansion ends in 7 consecutive 7’s. Then (10) tells us that a table of primes will some- where contain a million consecutive entries, each of which ends in the digits 7,777,777. 3 The Riemann Zeta-Function In the theory of the zeta-function it is customary to use the variable s = σ + it ∈ C. One then defines the complex exponential n−s := exp(−s log n), with log n ∈ R. The Riemann Zeta-function is then ζ(s) := ∞ n=1 n−s , σ 1. (3.1) The sum is absolutely convergent for σ 1, and for fixed δ 0 it is uniformly convergent for σ ≥ 1 + δ. It follows that ζ(s) is holomorphic for σ 1. The function is connected to the primes as follows. Theorem 6. (The Euler Product.) If σ 1 then we have ζ(s) = p (1 − p−s )−1 , where p runs over all primes, and the product is absolutely convergent.
- 22. 8 D. R. Heath-Brown This result is, philosophically, at the heart of the theory. It relates a sum over all positive integers to a product over primes. Thus it relates the additive structure, in which successive positive integers are generated by adding 1, to the multiplicative structure. Moreover we shall see in the proof that the fact that the sum and the product are equal is exactly an expression of the Fundamental Theorem of Arithmetic. To prove the result consider the finite product p≤X (1 − p−s )−1 . Since σ 1 we have |p−s | p−1 1, whence we can expand (1 − p−s )−1 as an absolutely convergent series 1 + p−s + p−2s + p−3s + . . .. We may multiply together a finite number of such series, and rearrange them, since we have absolute convergence. This yields p≤X (1 − p−s )−1 = ∞ n=1 aX (n) ns , where the coefficient aX (n) is the number of ways of writing n in the form n = pe1 1 pe2 2 . . . per r with p1 p2 . . . pr ≤ X. By the Fundamental Theorem of Arithmetic we have aX (n) = 0 or 1, and if n ≤ X we will have aX (n) = 1. It follows that | ∞ n=1 n−s − ∞ n=1 aX (n) ns | ≤ nX | 1 ns | = nX 1 nσ . As X → ∞ this final sum must tend to zero, since the infinite sum ∞ n=1 n−σ converges. We therefore deduce that if σ 1, then lim X→∞ p≤X (1 − p−s )−1 = ∞ n=1 1 ns , as required. Of course the product is absolutely convergent, as one may see by taking s = σ. One important deduction from the Euler product identity comes from tak- ing logarithms and differentiating termwise. This can be justified by the local uniform convergence of the resulting series. Corollary 1. We have − ζ ζ (s) = ∞ n=2 Λ(n) ns , (σ 1), (3.2) where Λ(n) = log p, n = pe , 0, otherwise. The function Λ(n) is known as the von Mangoldt function.
- 23. Prime number theory and the Riemann zeta-function 9 4 The Analytic Continuation and Functional Equation of ζ(s) Our definition only gives a meaning to ζ(s) when σ 1. We now seek to extend the definition to all s ∈ C. The key tool is the Poisson Summation Formula . Theorem 7. (The Poisson Summation Formula.) Suppose that f : R → R is twice differentiable and that f, f and f are all integrable over R. Define the Fourier transform by ˆ f(t) := ∞ −∞ f(x)e−2πitx dx. Then ∞ −∞ f(n) = ∞ −∞ ˆ f(n), both sides converging absolutely. There are weaker conditions under which this holds, but the above more than suffices for our application. The reader should note that there are a number of conventions in use for defining the Fourier transform, but the one used here is the most appropriate for number theoretic purposes. The proof (see Rademacher [12, page 71], for example) uses harmonic anal- ysis on R+ . Thus it depends only on the additive structure and not on the multiplicative structure. If we apply the theorem to f(x) = exp{−x2 πv}, which certainly fulfils the conditions, we have ˆ f(n) = ∞ −∞ e−x2 πv e−2πinx dx = ∞ −∞ e−πv(x+in/v)2 e−πn2 /v dx = e−πn2 /v ∞ −∞ e−πvy2 dy = 1 √ v e−πn2 /v , providing that v is real and positive. Thus if we define θ(v) := ∞ −∞ exp(−πn2 v), then the Poisson Summation Formula leads to the transformation formula θ(v) = 1 √ v θ(1/v).
- 24. 10 D. R. Heath-Brown The function θ(v) is a theta-function, and is an example of a modular form. It is the fact that θ(v) not only satisfies the above transformation formula when v goes to 1/v but is also periodic, that makes θ(v) a modular form. The “Langlands Philosophy” says that all reasonable generalizations of the Riemann Zeta-function are related to modular forms, in a suitably generalized sense. We are now ready to consider ζ(s), but first we introduce the function ψ(v) = ∞ n=1 e−n2 πv , (4.1) so that ψ(v) = (θ(v) − 1)/2 and 2ψ(v) + 1 = 1 √ v {2ψ( 1 v ) + 1}. (4.2) We proceed to compute that, if σ 1, then ∞ 0 xs/2−1 ψ(x)dx = ∞ n=1 ∞ 0 xs/2−1 e−n2 πx dx = ∞ n=1 1 (n2π)s/2 ∞ 0 ys/2−1 e−y dy = ∞ n=1 1 (n2π)s/2 Γ( s 2 ) = ζ(s)π−s/2 Γ( s 2 ), on substituting y = n2 πx. The interchange of summation and integration is justified by the absolute convergence of the resulting sum. We now split the range of integration in the original integral, and apply the transformation formula (4.2). For σ 1 we obtain the expression ζ(s)π−s/2 Γ( s 2 ) = ∞ 1 xs/2−1 ψ(x)dx + 1 0 xs/2−1 ψ(x)dx = ∞ 1 xs/2−1 ψ(x)dx + 1 0 xs/2−1 { 1 √ x ψ( 1 x ) + 1 2 √ x − 1 2 }dx = ∞ 1 xs/2−1 ψ(x)dx + 1 0 xs/2−3/2 ψ( 1 x )dx + 1 s − 1 − 1 s = ∞ 1 xs/2−1 ψ(x)dx + ∞ 1 y(1−s)/2−1 ψ(y)dy − 1 s(1 − s) , where we have substituted y for 1/x in the final integral. We therefore conclude that ζ(s)π−s/2 Γ( s 2 ) = ∞ 1 {xs/2−1 + x(1−s)/2−1 }ψ(x)dx − 1 s(1 − s) , (4.3)
- 25. Prime number theory and the Riemann zeta-function 11 whenever σ 1. However the right-hand side is meaningful for all values s ∈ C − {0, 1}, since the integral converges by virtue of the exponential decay of ψ(x). We may therefore use the above expression to define ζ(s) for all s ∈ C − {0, 1}, on noting that the factor π−s/2 Γ(s/2) never vanishes. Indeed, since Γ(s/2)−1 has a zero at s = 0 we see that the resulting expression for ζ(s) is regular at s = 0. Finally we observe that the right-hand side of (4.3) is invariant on substituting s for 1 − s. We are therefore led to the the following conclusion. Theorem 8. (Analytic Continuation and Functional Equation.) The function ζ(s) has an analytic continuation to C, and is regular apart from a simple pole at s = 1, with residue 1. Moreover π−s/2 Γ( s 2 )ζ(s) = π−(1−s)/2 Γ( 1 − s 2 )ζ(1 − s). Furthermore, if a ≤ σ ≤ b and |t| ≥ 1, then π−s/2 Γ(s 2 )ζ(s) is bounded in terms of a and b. To prove the last statement in the theorem we merely observe that |π−s/2 Γ( s 2 )ζ(s)| ≤ 1 + ∞ 1 (xb/2−1 + x(1−a)/2−1 )ψ(x)dx. 5 Zeros of ζ(s) It is convenient to define ξ(s) = 1 2 s(s − 1)π−s/2 Γ( s 2 )ζ(s) = (s − 1)π−s/2 Γ(1 + s 2 )ζ(s), (5.1) so that ξ(s) is entire. The functional equation then takes the form ξ(s) = ξ(1 − s). It is clear from (3.2) that ζ(s) can have no zeros for σ 1, since the series converges. Since 1/Γ(z) is entire, the function Γ(s/2) is non-vanishing, so that ξ(s) also has no zeros in σ 1. Thus, by the functional equation, the zeros of ξ(s) are confined to the “critical strip” 0 ≤ σ ≤ 1. Moreover any zero of ζ(s) must either be a zero of ξ(s), or a pole of Γ(s/2). We then see that the zeros of ζ(s) lie in the critical strip, with the exception of the “trivial zeros” at s = −2, −4, −6, . . . corresponding to poles of Γ(s/2). We may also observe that if ρ is a zero of ξ(s) then, by the functional equation, so is 1 − ρ. Moreover, since ξ(s) = ξ(s), we deduce that ρ and 1 − ρ are also zeros. Thus the zeros are symmetrically arranged about the real axis, and also about the “critical line” given by σ = 1/2. With this picture in mind we mention the following important conjectures. Conjecture 4. (The Riemann Hypothesis.) We have σ = 1/2 for all zeros of ξ(s). Conjecture 5. All zeros of ξ(s) are simple. In the absence of a proof of Conjecture 5 we adopt the convention that in any sum or product over zeros, we shall count them according to multiplicity.
- 26. 12 D. R. Heath-Brown 6 The Product Formula There is a useful product formula for ξ(s), due to Hadamard. In general we have the following result, for which see Davenport [3, Chapter 11] for example. Theorem 9. Let f(z) be an entire function with f(0) = 0, and suppose that there are constants A 0 and θ 2 such that f(z) = O(exp(A|z|θ )) for all complex z. Then there are constants α and β such that f(z) = eα+βz ∞ n=1 {(1 − z zn )ez/zn }, where zn runs over the zeros of f(z) counted with multiplicity. The infinite sum ∞ n=1 |zn|−2 converges, so that the product above is absolutely and uniformly convergent in any compact set which includes none of the zeros. We can apply this to ξ(s), since it is apparent from Theorem 8, together with the definition (5.1) that ξ(0) = ξ(1) = 1 2 π−1/2 Γ( 1 2 )Res{ζ(s); s = 1} = 1 2 . For σ ≥ 2 one has ζ(s) = O(1) directly from the series (3.1), while Stir- ling’s approximation yields Γ(s/2) = O(exp(|s| log |s|)). It follows that ξ(s) = O(exp(|s| log |s|)) whenever σ ≥ 2. Moreover, when 1 2 ≤ σ ≤ 2 one sees from Theorem 8 that ξ(s) is bounded. Thus, using the functional equation, we can deduce that ξ(s) = O(exp(|s| log |s|)) for all s with |s| ≥ 2. We may therefore deduce from Theorem 9 that ξ(s) = eα+βs ρ {(1 − s ρ )es/ρ }, where ρ runs over the zeros of ξ(s). Thus, with appropriate branches of the logarithms, we have log ξ(s) = α + βs + ρ {log(1 − s ρ ) + s ρ }. We can then differentiate termwise to deduce that ξ ξ (s) = β + ρ { 1 s − ρ + 1 ρ }, the termwise differentiation being justified by the local uniform convergence of the resulting sum. We therefore deduce that ζ ζ (s) = β − 1 s − 1 + 1 2 log π − 1 2 Γ Γ ( s 2 + 1) + ρ { 1 s − ρ + 1 ρ }, (6.1) where, as ever, ρ runs over the zeros of ξ counted according to multiplicity. In fact, on taking s → 1, one can show that β = − 1 2 γ − 1 − 1 2 log 4π, where γ is Euler’s constant. However we shall make no use of this fact.
- 27. Prime number theory and the Riemann zeta-function 13 7 The Functions N(T) and S(T) We shall now investigate the frequency of the zeros ρ. We define N(T) = #{ρ = β + iγ : 0 ≤ β ≤ 1, 0 ≤ γ ≤ T}. The notation β = (ρ), γ = (ρ) is standard. In fact one can easily show that ψ(x) (2 √ x)−1 , whence (4.3) suffices to prove that ζ(s) 0 for real s ∈ (0, 1). Thus we have γ 0 for any zero counted by N(T). The first result we shall prove is the following. Theorem 10. If T is not the ordinate of a zero, then N(T) = T 2π log T 2π − T 2π + 7 8 + S(T) + O(1/T), where S(T) = 1 π arg ζ( 1 2 + iT), is defined by continuous variation along the line segments from 2 to 2 + iT to 1 2 + iT. We shall evaluate N(T) using the Principle of the Argument, which shows that N(T) = 1 2π ∆R arg ξ(s), providing that T is not the ordinate of any zero. Here R is the rectangular path joining 2, 2 + iT, −1 + iT, and −1. To calculate ∆R arg ξ(s) one starts with any branch of arg ξ(s) and allows it to vary continuously around the path. Then ∆R arg ξ(s) is the increase in arg ξ(s) along the path. Our assumption about T ensures that ξ(s) does not vanish on R. Now ξ(s) = ξ(1 − s) and ξ(1 − s) = ξ(1 − s), whence ξ(1 2 + a + ib) is conjugate to ξ(1 2 − a + ib). (In particular this shows that ξ(1 2 + it) is always real.) It follows that ∆Rξ(s) = 2∆P ξ(s), where P is the path 1 2 → 2 → 2 + iT → 1 2 + iT. On the first line segment ξ(s) is real and strictly positive, so that the contribution to ∆P ξ(s) is zero. Let L be the remaining path 2 → 2 + iT → 1 2 + iT. Then ∆Lξ(s) = ∆L{arg(s − 1)π−s/2 Γ( s 2 + 1)} + ∆L arg ζ(s). Now on L the function s − 1 goes from 1 to −1 2 + iT, whence ∆L arg(s − 1) = arg(− 1 2 + iT) = π 2 + O(T−1 ). We also have arg π−s/2 = log π−s/2 = (− s 2 log π),
- 28. 14 D. R. Heath-Brown so that arg π−s/2 goes from 0 to −(T log π)/2 and ∆L arg π−s/2 = − T 2 log π. Finally, Stirling’s formula yields log Γ(z) = (z − 1 2 ) log z − z + 1 2 log(2π) + O(|z|−1 ), (| arg(z)| ≤ π − δ), (7.1) whence ∆L arg Γ( s 2 + 1) = log Γ( 1 2 + iT 2 + 1) = {( 3 4 + i T 2 ) log( 5 4 + i T 2 ) − ( 5 4 + i T 2 ) + 1 2 log(2π)} +O(1/T) = T 2 log T 2 − T 2 + 3π 8 + O(1/T), since log( 5 4 + i T 2 ) = log T 2 + i π 2 + O(1/T). These results suffice for Theorem 10 We now need to know about S(T). Here we show the following. Theorem 11. We have S(T) = O(log T). Corollary 2. (The Riemann – von Mangoldt Formula). We have N(T) = T 2π log T 2π − T 2π + O(log T). We start the proof by taking s = 2 + iT in (3.2) and noting that | ζ ζ (s)| ≤ ∞ n=2 Λ(n) n2 = O(1). Thus the partial fraction decomposition (6.1) yields ρ { 1 2 + iT − ρ + 1 ρ } = 1 2 Γ Γ (2 + iT 2 ) + O(1). We may differentiate (7.1), using Cauchy’s formula for the first derivative, to produce Γ Γ (z) = log z + O(1), (| arg(z)| ≤ π − δ), (7.2) and then deduce that ρ { 1 2 + iT − ρ + 1 ρ } = O(log(2 + T)). (7.3)
- 29. Prime number theory and the Riemann zeta-function 15 We have only assumed here that T ≥ 0, not that T ≥ 2. In order to get the correct order estimate when 0 ≤ T ≤ 2 we have therefore written O(log(2+T)), which is O(1) for 0 ≤ T ≤ 2. Setting ρ = β + iγ we now have 1 2 + iT − ρ = 2 − β (2 − β)2 + (T − γ)2 ≥ 1 4 + (T − γ)2 and 1 ρ = β β2 + γ2 ≥ 0, since 0 ≤ β ≤ 1. We therefore produce the useful estimate ρ 1 4 + (T − γ)2 = O(log(2 + T)), (7.4) which implies in particular that #{ρ : T − 1 ≤ γ ≤ T + 1} = O(log(2 + T)). (7.5) We now apply (6.1) with s = σ + iT and 0 ≤ σ ≤ 2, and subtract (7.3) from it to produce ζ ζ (σ + iT) = − 1 σ + iT − 1 + ρ { 1 σ + iT − ρ − 1 2 + iT − ρ } + O(log(2 + T)). Terms with |γ − T| 1 have | 1 σ + iT − ρ − 1 2 + iT − ρ | = | 2 − σ (σ + iT − ρ)(2 + iT − ρ) | ≤ 2 |γ − T|.|γ − T| ≤ 2 1 5 {4 + (T − γ)2} . Thus (7.4) implies that ρ: |γ−T |1 { 1 σ + iT − ρ − 1 2 + iT − ρ } = O(log(2 + T)), and hence that ζ ζ (σ + iT) = − 1 σ + iT − 1 + ρ: |γ−T |≤1 { 1 σ + iT − ρ − 1 2 + iT − ρ } +O(log(2 + T)).
- 30. 16 D. R. Heath-Brown However we also have | 1 2 + iT − ρ | ≤ 1 2 − β ≤ 1, whence (7.5) produces ρ: |γ−T |≤1 1 2 + iT − ρ = O(log(2 + T)). We therefore deduce the following estimate. Lemma 1. For 0 ≤ σ ≤ 2 and T ≥ 0 we have ζ ζ (σ + iT) = − 1 σ + iT − 1 + ρ: |γ−T |≤1 1 σ + iT − ρ + O(log(2 + T)). We are now ready to complete our estimation of S(T). Taking T ≥ 2, we have arg ζ( 1 2 + iT) = log ζ( 1 2 + iT) = 1/2+iT 2 ζ ζ (s)ds, the path of integration consisting of the line segments from 2 to 2 + iT and from 2 + iT to 1/2 + iT. Along the first of these we use the formula (3.2), which yields 2+iT 2 ζ ζ (s)ds = ∞ n=2 Λ(n) ns log n 2+iT 2 = O(1). For the remaining range we use Lemma 1, which produces 1/2+iT 2+iT ζ ζ (s)ds = ρ: |γ−T |≤1 1/2+iT 2+iT ds s − ρ + O(log T) = ρ: |γ−T |≤1 {log( 1 2 + iT − ρ) − log(2 + iT − ρ)} +O(log T) = ρ: |γ−T |≤1 {arg( 1 2 + iT − ρ) − arg(2 + iT − ρ)} +O(log T) = ρ: |γ−T |≤1 O(1) + O(log T) = O(log T), by (7.5). This suffices for the proof of Theorem 11.
- 31. Prime number theory and the Riemann zeta-function 17 8 The Non-Vanishing of ζ(s) on σ = 1 So far we know only that the non-trivial zeros of ζ(s) lie in the critical strip 0 ≤ σ ≤ 1. Qualitatively the only further information we have is that there are no zeros on the boundary of this strip. Theorem 12. (Hadamard and de la Vallée Poussin, independently, 1896.) We have ζ(1 + it) = 0, for all real t. This result was the key to the proof of the Prime Number Theorem. Quan- titatively one can say a little more. Theorem 13. (De la Vallée Poussin.) There is a positive absolute constant c such that for any T ≥ 2 there are no zeros of ζ(s) in the region σ ≥ 1 − c log T , |t| ≤ T. In fact, with much more work, one can replace the function c/ log T by one that tends to zero slightly more slowly, but that will not concern us here. The proof of Theorem 13 uses the following simple fact. Lemma 2. For any real θ we have 3 + 4 cos θ + cos 2θ ≥ 0. This is obvious, since 3 + 4 cos θ + cos 2θ = 2{1 + cos θ}2 . We now use the identity (3.2) to show that −3 ζ ζ (σ) − 4 ζ ζ (σ + it) − ζ ζ (σ + 2it) = ∞ n=2 Λ(n) nσ {3 + 4 cos(t log n) + cos(2t log n)} ≥ 0, for σ 1. When 1 σ ≤ 2 we have − ζ ζ (σ) = 1 σ − 1 + O(1), from the Laurent expansion around the pole at s = 1. For the remaining two terms we use Lemma 1, to deduce that 3 σ − 1 + O(1) − 4 ρ: |γ−t|≤1 1 σ + it − ρ − ρ: |γ−2t|≤1 1 σ + 2it − ρ + O(log T)
- 32. 18 D. R. Heath-Brown ≥ 0 for 1 σ ≤ 2, T ≥ 2, and |t| ≤ T. Suppose we have a zero ρ0 = β0 + iγ0, say, with 0 ≤ γ0 ≤ T. Set t = γ0. We then observe that for any zero we have 1 σ + it − ρ = σ − β (σ − β)2 + (t − γ)2 ≥ 0, since σ 1 ≥ β, and similarly 1 σ + 2it − ρ ≥ 0. We can therefore drop all terms from the two sums above, with the exception of the term corresponding to ρ = ρ0, to deduce that 4 σ − β0 ≤ 3 σ − 1 + O(log T). Suppose that the constant implied by the O(. . .) notation is c0. This is just a numerical value that one could calculate with a little effort. Then 4 σ − β0 ≤ 3 σ − 1 + c0 log T whenever 1 σ ≤ 2. If β0 = 1 we get an immediate contradiction by choosing σ = 1 + (2c0 log T)−1 . If β0 3/4 the result of Theorem 13 is immediate. For the remaining range of β0 we choose σ = 1 + 4(1 − β0), which will show that 4 5(1 − β0) ≤ 3 4(1 − β0) + c0 log T. Thus 1 20(1 − β0) ≤ c0 log T, and hence 1 − β0 ≥ 1 20c0 log T . This completes the proof of Theorem 13. The reader should observe that the key feature of the inequality given in Lemma 2 is that the coefficients are non-negative, and that the coefficient of cos θ is strictly greater than the constant term. In particular, the inequality 1 + cos θ ≥ 0 just fails to work. Theorem 13 has a useful corollary.
- 33. Prime number theory and the Riemann zeta-function 19 Corollary 3. Let c be as in Theorem 13, and let T ≥ 2. Then if 1 − c 2 log T ≤ σ ≤ 2 and |t| ≤ T, we have ζ ζ (σ + it) = − 1 σ + it − 1 + O(log2 T). For the proof we use Lemma 1. The sum over zeros has O(log T) terms, by (7.5), and each term is O(log T), since σ − β ≥ c 2 log T , by Theorem 13. 9 Proof of the Prime Number Theorem Since our argument is based on the formula (3.2), it is natural to work with Λ(n). We define ψ(x) = n≤x Λ(n) = pk ≤x log p. (9.1) This is not the same function as that defined in (4.1)! Our sum ψ(x) is related to π(x) in the following lemma. Lemma 3. For x ≥ 2 we have π(x) = ψ(x) log x + x 2 ψ(t) t log2 t dt + O(x1/2 ). For the proof we begin by setting θ(x) = p≤x log p. Then x 2 θ(t) t log2 t dt = x 2 p≤t log p t log2 t dt = p≤x x p log p t log2 t dt = p≤x − log p log t x p = π(x) − θ(x) log x ,
- 34. 20 D. R. Heath-Brown so that π(x) = θ(x) log x + x 2 θ(t) t log2 t dt. (9.2) However it is clear that terms in (9.1) with k ≥ 2 have p ≤ x1/2 , and there are at most x1/2 such p. Moreover k ≤ log x/ log p, whence the total contribution from terms with k ≥ 2 is O(x1/2 log x). Thus ψ(x) = θ(x) + O(x1/2 log x). If we substitute this into (9.2) the required result follows. We will use contour integration to relate ψ(x) to ζ (s)/ζ(s). This will be done via the following result. Lemma 4. Let y 0, c 1 and T ≥ 1. Define I(y, T) = 1 2πi c+iT c−iT ys s ds. Then I(y, T) = 0, 0 y 1 1, y 1 + O( yc T| log y| ). When 0 y 1 we replace the path of integration by the line segments c − iT → N − iT → N + iT → c + iT, and let N → ∞. Then N+iT N−iT ys s ds → 0, while N−iT c−iT ys s ds = O( N c yσ T dσ) = O( yc T| log y| ), and similarly for the integral from N + iT to c + iT. It follows that I(y, T) = O( yc T| log y| ) for 0 y 1. The case y 1 can be treated analogously, using the path c − iT → −N − iT → −N + iT → c + iT. However in this case we pass a pole at s = 0, with residue 1, and this produces the corresponding main term for I(y, T). We can now give our formula for ψ(x). Theorem 14. For x − 1 2 ∈ N, α = 1 + 1/ log x and T ≥ 1 we have ψ(x) = 1 2πi α+iT α−iT {− ζ ζ (s)} xs s ds + O( x log2 x T ).
- 35. Prime number theory and the Riemann zeta-function 21 For the proof we integrate termwise to get 1 2πi α+iT α−iT {− ζ ζ (s)} xs s ds = ∞ n=2 Λ(n)I( x n , T) = n≤x Λ(n) + O( ∞ n=2 Λ(n)( x n )α 1 T| log x/n| ). Since we are taking x − 1 2 ∈ N the case x/n = 1 does not occur. In the error sum those terms with n ≤ x/2 or n ≥ 2x have | log x/n| ≥ log 2. Such terms therefore contribute O( ∞ n=2 Λ(n) xα Tnα ) = O( xα T | ζ ζ (α)|) = O( x1+1/ log x T 1 α − 1 ) = O( x log x T ). When x/2 n 2x we have | log x/n| ≥ 1 2 |x − n| x and Λ(n)( x n )α = O(log x). These terms therefore contribute x/2n2x O( x log x T|x − n| ) = O( x log2 x T ) on bearing in mind that x − 1 2 ∈ N. The theorem now follows. We are now ready to prove the following major result. Theorem 15. There is a positive constant c0 such that ψ(x) = x + O(x exp{−c0 log x}) (9.3) for all x ≥ 2. Moreover we have π(x) = Li(x) + O(x exp{−c0 log x}) for all x ≥ 2. The error terms here can be improved slightly, but with considerably more work.
- 36. 22 D. R. Heath-Brown It clearly suffices to consider the case in which x − 1 2 ∈ N. To prove the result we set µ = 1 − c 2 log T , T ≥ 2, as in Lemma 3, and replace the line of integration in Theorem 14 by the path α − iT → µ − iT → µ + iT → α + iT. The integrand has a pole at s = 1 with residue x, arising from the pole of ζ(s), but no other singularities, by virtue of Theorem 13. On the new path of integration Lemma 3 shows that ζ ζ (s) = O(log2 T). We therefore deduce that ψ(x) = x + O( x log2 x T ) + O( α µ log2 T T xσ dσ) + O( T −T log2 T |µ + it| xµ dt), where the first error integral corresponds to the line segments α−iT → µ−iT and µ + iT → α + iT, and the second error integral to the segment µ − iT → µ + iT. These integrals are readily estimated to yield ψ(x) = x + O( x log2 x T ) + O( log2 T T xα ) + O(xµ log3 T). Of course xα = O(x) here. Thus if T ≤ x we merely get ψ(x) = x + O(x log3 x{ 1 T + xµ−1 }). We now choose T = exp{ log x}, whence ψ(x) = x + O(x(log x)3 exp{− min(1, c 2 ) log x}). We may therefore choose any positive constant c0 min(1, c 2 ) in Theorem 15. This establishes (9.3). To prove the remaining assertion, it suffices to insert (9.3) into Lemma 3. Finally we should stress that the success of this argument depends on being able to take µ 1, since there is an error term which is essentially of order xµ . Thus it is crucial that we should at least know that ζ(1 + it) = 0. If we assume the Riemann Hypothesis, then we may take any µ 1 2 in the above analysis. This leads to the following estimates. Theorem 16. On the Riemann Hypothesis we have ψ(x) = x + O(xθ ) and π(x) = Li(x) + O(xθ ) for any θ 1 2 and all x ≥ 2. One cannot reduce the exponent below 1/2, since there is a genuine con- tribution to π(x) arising from the zeros of ζ(s).
- 37. Prime number theory and the Riemann zeta-function 23 10 Explicit Formulae In this section we shall argue somewhat informally, and present results without proof. If f : (0, ∞) → C we define the Mellin transform of f to be the function F(s) := ∞ 0 f(x)xs−1 dx. By a suitable change of variables one sees that this is essentially a form of Fourier transform. Indeed all the properties of Mellin transforms can readily be translated from standard results about Fourier transforms. In particular, under suitable conditions one has an inversion formula f(x) = 1 2πi σ+i∞ σ−i∞ F(s)x−s ds. Arguing purely formally one then has ∞ n=2 Λ(n)f(n) = ∞ n=2 Λ(n) 1 2πi 2+i∞ 2−i∞ F(s)n−s ds = 1 2πi 2+i∞ 2−i∞ {− ζ ζ (s)}F(s)ds. If one now moves the line of integration to (s) = −N one passes poles at s = 1 and at s = ρ for every non-trivial zero ρ, as well as at the trivial zeros −2n. Under suitable conditions the integral along (s) = −N will tend to 0 as N → ∞. This argument leads to the following result. Theorem 17. Suppose that f ∈ C2 (0, ∞) and that supp(f) ⊆ [1, X] for some X. Then ∞ n=2 Λ(n)f(n) = F(1) − ρ F(ρ) − ∞ n=1 F(−2n). One can prove such results subject to weaker conditions on f. If x is given, and f(t) = 1, t ≤ x 0, t x, then the conditions above are certainly not satisfied, but we have the following related result. Theorem 18. (The Explicit Formula.) Let x ≥ T ≥ 2. Then ψ(x) = x − ρ: |γ|≤T xρ ρ + O( x log2 x T ).
- 38. 24 D. R. Heath-Brown For a proof of this see Davenport [3, Chapter 17], for example. There are variants of this result containing a sum over all zeros, and with no error term, but the above is usually more useful. The explicit formula shows exactly how the zeros influence the behaviour of ψ(x), and hence of π(x). The connection between zeros and primes is even more clearly shown by the following result of Landau. Theorem 19. For fixed positive real x define Λ(x) = 0 if x ∈ N and Λ(x) = Λ(n) if x = n ∈ N. Then Λ(x) = − 2π T ρ: 0γ≤T xρ + Ox( log T T ), where Ox(. . .) indicates that the implied constant may depend on x. This result shows that the zeros precisely determine the primes. Thus, for example, one can reformulate the conjecture (1.1) as a statement about the zeros of the zeta-function. All the unevenness of the primes, for example the behaviour described by Theorem 5, is encoded in the zeros of the zeta-function. It therefore seems reasonable to expect that the zeros themselves should have corresponding unevenness. 11 Dirichlet Characters We now turn to the simplest type of generalization of the Riemann Zeta- function, namely the Dirichlet L-functions. In the remainder of these notes we shall omit most of the proofs, being content merely to describe what can be proved. A straightforward example of a Dirichlet L-function is provided by the infinite series 1 − 1 3s + 1 5s − 1 7s + 1 9s − 1 11s + . . . . (11.1) We first need to describe the coefficients which arise. Definition . Let q ∈ N. A “(Dirichlet) character χ to modulus q” is a function χ : Z → C such that (i) χ(mn) = χ(m)χ(n) for all m, n ∈ Z; (ii) χ(n) has period q; (iii) χ(n) = 0 whenever (n, q) = 1; and (iv) χ(1) = 1.
- 39. Prime number theory and the Riemann zeta-function 25 Part (iv) of the definition is necessary merely to rule out the possibility that χ is identically zero. As an example we can take the function χ(n) = ⎧ ⎨ ⎩ 1, n ≡ 1 (mod 4), −1, n ≡ 3 (mod 4), 0, n ≡ 0 (mod 2). (11.2) This is a character modulo 4, and is the one generating the series (11.1). A second example is the function χ0(n) := 1, (n, q) = 1, 0, (n, q) = 1. This produces a character for every modulus q, known as the principal char- acter modulo q. A number of key facts are gathered together in the following theorem. Theorem 20. (i) We have |χ(n)| = 1 for every n coprime to q. (ii) If χ1 and χ2 are two characters to modulus q, then so is χ1χ2, where we define χ1χ2(n) = χ1(n)χ2(n). (iii) There are exactly ϕ(q) different characters to modulus q. (iv) If n ≡ 1(mod q) then there is at least one character χ modulo q for which χ(n) = 1. In part (iii) the function ϕ(q) is the number of positive integers n ≤ q for which n and q are coprime. To prove part (i) we note that the sequence nk (mod q) must eventually repeat when k runs through N. Thus there exist k j with χ(nk ) = χ(nj ), and hence χ(n)k = χ(n)j . Since n is coprime to q we have χ(n) = 0, so that χ(n)j−k = 1. Part (ii) of the theorem is obvious, but parts (iii) and (iv) are harder, and we refer the reader to Davenport [3, Chapter 4] for the details. As an example of (iii) we note that ϕ(4) = 2, and we have already found two characters modulo 4. There are no others. One further fact may elucidate the situation. Consider a general finite abelian group G. In our case we will have G = (Z/qZ)× . Thus G will consist of those residue classes n mod q for which (n, q) = 1, with the multiplica- tion operation. Define G to be the group of homomorphisms θ : G → C× , where the group action is given by (θ1θ2)(g) := θ1(g)θ2(g). In our case these homomorphisms are, in effect, the characters. Then the groups G and G are isomorphic, and part (iii) above is an immediate consequence. The details can be found in Ireland and Rosen [7, pages 253 and 254], for example. Indeed
- 40. 26 D. R. Heath-Brown there is a duality between G and G. The isomorphism between them is not “natural”, but there is a natural isomorphism G G. There are two orthogonality relations satisfied by the characters to a given modulus q. The first of these is the following. Theorem 21. If a and q are coprime then S := χ(mod q) χ(n)χ(a) = ϕ(q), n ≡ a (mod q), 0, n ≡ a (mod q). When n ≡ a(mod q) this is immediate since then χ(n)χ(a) = 1 for all χ. In the remaining case, choose an element b with ab ≡ 1(mod q). By (iv) of Theorem 20 there exists a character χ1 such that χ1(nb) = 1. Then χ1(nb)S = χ(mod q) χ1(n)χ(n)χ1(b)χ(a). However χ1(b)χ1(a) = χ1(ab) = χ1(1) = 1, whence χ1(b) = χ1(a). We therefore deduce that χ1(nb)S = χ(mod q) χ1(n)χ(n)χ1(a)χ(a) = χ(mod q) χ1χ(n)χ1χ(a). As χ runs over the complete set of characters to modulus q the product χ1χ does as well, since χ1χ = χ1χ implies χ = χ . Thus χ(mod q) χ1χ(n)χ1χ(a) = S and hence χ1(nb)S = S. Since χ1(nb) = 1 we deduce that S = 0, as required. The second orthogonality relation is the following. Theorem 22. If χ = χ0 then q n=1 χ(n) = 0. The proof is analogous to the previous result, and is based on the obvious fact that if χ = χ0 then there is some integer n coprime to q such that χ(n) = 1. The details are left as an exercise for the reader. If q has a factor r and χ is a character modulo r we can define the character ψ modulo q which is “induced by” χ. This is done by setting ψ(n) = χ(n), (n, q) = 1, 0, (n, q) = 1.
- 41. Prime number theory and the Riemann zeta-function 27 For example, we may take χ to be the character modulo 4 given by (11.2). Then if q = 12 we induce a character ψ modulo 12, as in the following table. 1 2 3 4 5 6 7 8 9 10 11 12 χ 1 0 -1 0 1 0 -1 0 1 0 -1 0 ψ 1 0 0 0 1 0 -1 0 0 0 -1 0 A character χ(mod q) which cannot be produced this way from some divisor r q is said to be “primitive”. The principal character is induced by the character χ0 (mod 1), that is to say by the character which is identically 1. If q is prime, then all the characters except for the principal character are primitive. In general there will be both primitive and imprimitive characters to each modulus. Imprimitive characters are a real nuisance!! 12 Dirichlet L-functions For any character χ to modulus q we will define the corresponding Dirichlet L-function by setting L(s, χ) = ∞ n=1 χ(n) ns , (σ 1). We content ourselves here with describing the key features of these func- tions, and refer the reader to Davenport [3], for example, for details. The sum is absolutely convergent for σ 1 and is locally uniformly con- vergent, so that L(s, χ) is holomorphic in this region. If χ is the principal character modulo q then the series fails to converge when σ ≤ 1. However for non-principal χ the series is conditionally convergent when σ 0, and the series defines a holomorphic function in this larger region. There is an Euler product identity L(s, χ) = p (1 − χ(p)p−s )−1 , (σ 1). This can be proved in the same way as for ζ(s) using the multiplicativity of the function χ. Suppose that χ is primitive, and that χ(−1) = 1. If we apply the Poisson summation formula to f(x) = e−(a+qx)2 πv/q , multiply the result by χ(a), and sum for 1 ≤ a ≤ q, we find that θ(v, χ) = τ(χ) √ q 1 √ v θ( 1 v , χ), where θ(v, χ) := ∞ n=−∞ χ(n)e−n2 πv/q
- 42. 28 D. R. Heath-Brown is a generalisation of the theta-function, and τ(χ) := q a=1 χ(a)e2πia/q is the Gauss sum. When χ is primitive and χ(−1) = −1 the function θ(v, χ) vanishes identi- cally. Instead we use θ1(v, χ) := ∞ n=−∞ nχ(n)e−n2 πv/q , for which one finds the analogous transformation formula θ1(v, χ) = iτ(χ) √ q 1 v3/2 θ1( 1 v , χ). These two transformation formulae then lead to the analytic continuation and functional equation for L(s, χ). The conclusion is that, if χ is primitive then L(s, χ) has an analytic continuation to the whole complex plane, and is regular everywhere, except when χ is identically 1, (in which case L(s, χ) is just the Riemann Zeta-function ζ(s)). Moreover, still assuming that χ is primitive, with modulus q, we set ξ(s, χ) = ( q π )(s+a)/2 Γ( s + a 2 )L(s, χ), where a = a(χ) := 0, χ(−1) = 1, 1, χ(−1) = −1. Then ξ(1 − s, χ) = ia q1/2 τ(χ) ξ(s, χ). Notice in particular that, unless the values taken by χ are all real, this func- tional equation relates L(s, χ) not to the same function at 1 − s but to the conjugate L-function, with character χ. It follows from the Euler product and the functional equation that there are no zeros of ξ(s, χ) outside the critical strip. The zeros will be symmetrically distributed about the critical line σ = 1/2, but unless χ is real they will not necessarily be symmetric about the real line. Hence in general it is appropriate to define N(T, χ) := #{ρ : ξ(ρ, χ) = 0 |γ| ≤ T}, counting zeros both above and below the real axis. We then have 1 2 N(T, χ) = T 2π log qT 2π − T 2π + O(log qT)
- 43. Prime number theory and the Riemann zeta-function 29 for T ≥ 2, which can be seen as an analogue of the Riemann – von Mangoldt formula. This shows in particular that the interval [T, T + 1] contains around 1 2π log qT 2π zeros, on average. The work on regions without zeros can be generalized, but there are serious problems with possible zeros on the real axis. Thus one can show that there is a constant c 0, which is independent of q, such that if T ≥ 2 then L(s, χ) has no zeros in the region σ ≥ 1 − c log qT , 0 |t| ≤ T. If χ is not a real-valued character then we can extend this result to the case t = 0, but is a significant open problem to deal with the case in which χ is real. However in many other important aspects techniques used for the Riemann Zeta-function can be successfully generalized to handle Dirichlet L-functions. References [1] R.C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II., Proc. London Math. Soc. (3), 83 (2001), 532-562. [2] J.-R. Chen, on the representation of a large even integer as a sum of a prime and a product of at most two primes, Kexue Tongbao, 17 (1966), 385-386. [3] H. Davenport, Multiplicative number theory, Graduate Texts in Mathe- matics, 74. (Springer-Verlag, New York-Berlin, 1980). [4] J.B. Friedlander and H. Iwaniec, The polynomial X2 + Y 4 captures its primes, Annals of Math. (2), 148 (1998), 945-1040. [5] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, (Oxford University Press, New York, 1979). [6] A.E. Ingham, On the difference between consecutive primes, Quart. J. Math. Oxford, 8 (1937), 255-266. [7] K. Ireland and M. Rosen, A classical introduction to modern number the- ory, Graduate Texts in Math., 84, (Springer, Heidelberg-New York, 1990). [8] H. Iwaniec, Almost-primes represented by quadratic polynomials, Invent. Math., 47 (1978), 171-188. [9] H. Maier, Primes in short intervals, Michigan Math. J., 32 (1985), 221- 225.
- 44. 30 D. R. Heath-Brown [10] H. Maier, Small differences between prime numbers, Michigan Math. J., 35 (1988), 323-344. [11] I.I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [f(n)], Mat. Sbornik N.S., 33(75) (1953), 559-566. [12] H. Rademacher, Topics in analytic number theory, Grundlehren math. Wiss., 169, (Springer, New York-Heidelberg, 1973). [13] R.A. Rankin, The difference between consecutive prime numbers, J. Lon- don Math. Soc., 13 (1938), 242-247. [14] J. Rivat and J. Wu, Prime numbers of the form [nc ], Glasg. Math. J., 43 (2001), 237-254. [15] A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid., 47, (1943), 87-105. [16] D.K.L. Shiu, Strings of congruent primes, J. London Math. Soc. (2), 61 (2000), 359-373. Mathematical Institute, 24-29, St. Giles’, Oxford OX1 3LB rhb@maths.ox.ac.uk
- 45. Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond Yan V. Fyodorov Abstract These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large ran- dom Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Uni- tary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in vari- ous regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal poly- nomials and characteristic polynomials of random matrices which is an active area of current research. 1 Preface Gaussian Ensembles of random Hermitian or real symmetric matrices always played a prominent role in the development and applications of Random Ma- trix Theory. Gaussian Ensembles are uniquely singled out by the fact that
- 46. 32 Yan V. Fyodorov they belong both to the family of invariant ensembles, and to the family of ensembles with independent, identically distributed (i.i.d) entries. In general, mathematical methods used to treat those two families are very different. In fact, all random matrix techniques and ideas can be most clearly and consistently introduced using Gaussian case as a paradigmatic example. In the present set of lectures we mainly concentrate on consequences of the invari- ance of the corresponding probability density function, leaving aside methods of exploiting statistical independence of matrix entries. Under these circum- stances the method of orthogonal polynomials is the most adequate one, and for the Gaussian case the relevant polynomials are Hermite polynomials. Being mostly interested in the limit of large matrix sizes we will spend a considerable amount of time investigating various asymptotic regimes of Hermite polyno- mials, since the latter are main building blocks of various correlation functions of interest. In the last part of our lecture course we will discuss why statistics of characteristic polynomials of random Hermitian matrices turns out to be interesting and informative to investigate, and will make a contact with recent results in the domain. The presentation is quite informal in the sense that I will not try to prove various statements in full rigor or generality. I rather attempt outlining the main concepts, ideas and techniques preferring a good illuminating example to a general proof. A much more rigorous and detailed exposition can be found in the cited literature. I will also frequently employ the symbol ∝. In the present set of lectures it always means that the expression following ∝ contains a multiplicative constant which is of secondary importance for our goals and can be restored when necessary. 2 Introduction In these lectures we use the symbol T to denote matrix or vector transposition and the asterisk ∗ to denote Hermitian conjugation. In the present section the bar z denotes complex conjugation. Let us start with a square complex matrix Ẑ of dimensions N × N, with complex entries zij = xij + iyij, 1 ≤ i, j ≤ N. Every such matrix can be conveniently looked at as a point in a 2N2 -dimensional Euclidean space with real Cartesian coordinates xij, yij, and the length element in this space is defined in a standard way as: (ds)2 = Tr dẐdẐ∗ = ij dzijdzij = ij (dx)2 ij + (dy)2 ij . (2.1) As is well-known (see e.g.[1]) any surface embedded in an Euclidean space inherits a natural Riemannian metric from the underlying Euclidean struc- ture. Namely, let the coordinates in a n−dimensional Euclidean space be (x1, . . . , xn), and let a k−dimensional surface embedded in this space be param- eterized in terms of coordinates (q1, . . . , qk), k ≤ n as xi = xi(q1, . . . , qk), i =
- 47. Introduction to the random matrix theory: Gaussian Unitary Ensemble and beyond 33 1, . . . n. Then the Riemannian metric gml = glm on the surface is defined from the Euclidean length element according to (ds)2 = n i=1 (dxi)2 = n i=1 k m=1 ∂xi ∂qm dqm 2 = k m,l=1 gmndqmdql. (2.2) Moreover, such a Riemannian metric induces the corresponding integration measure on the surface, with the volume element given by dµ = |g|dq1 . . . dqk, g = det (gml)k l,m=1. (2.3) For k = n these are just the familiar formulae for the lengths and volume associated with change of coordinates in an Euclidean space. For example, for n = 2 we can pass from Cartesian coordinates −∞ x, y ∞ to polar coordinates r 0, 0 ≤ θ 2π by x = r cos θ, y = r sin θ, so that dx = dr cos θ − r sin θdθ, dy = dr sin θ + r cos θdθ, and the Riemannian metric is defined by (ds)2 = (dx)2 +(dy)2 = (dr)2 +r2 (dθ)2 . We find that g11 = 1, g12 = g21 = 0, g22 = r2 , and the volume element of the integration measure in the new coordinates is dµ = rdrdθ; as it should be. As the simplest example of a “surface” with k n = 2 embedded in such a two-dimensional space we consider a circle r = R = const. We immediately see that the length element (ds)2 restricted to this “surface” is (ds)2 = R2 (dθ)2 , so that g11 = R2 , and the integration measure induced on the surface is correspondingly dµ = Rdθ. The “surface” integration then gives the total “volume” of the embedded surface (i.e. circle length 2πR). z y x θ φ Figure 1: The spherical coordinates for a two dimensional sphere in the three- dimensional Euclidean space. Similarly, we can consider a two-dimensional (k = 2) sphere R2 = x2 +y2 + z2 embedded in a three-dimensional Euclidean space (n = 3) with coordinates x, y, z and length element (ds)2 = (dx)2 + (dy)2 + (dz)2 . A natural param- eterization of the points on the sphere is possible in terms of the spherical coordinates φ, θ (see Fig. 1) x = R sin θ cos φ, y = R sin θ sin φ, z = R cos θ; 0 ≤ θ ≤ π, 0 ≤ φ 2π,
- 48. 34 Yan V. Fyodorov which results in (ds)2 = R2 (dθ)2 + R2 sin2 θ(dφ)2 . Hence the matrix elements of the metric are g11 = R2 , g12 = g21 = 0, g22 = R2 sin2 θ, and the cor- responding “volume element” on the sphere is the familiar elementary area dµ = R2 sin θdθdφ. As a less trivial example to be used later on consider a 2−dimensional manifold formed by 2 × 2 unitary matrices Û embedded in the 8 dimensional Euclidean space of Gl(2; C) matrices. Every such matrix can be represented as the product of a matrix Ûc from the coset space U(2)/U(1) × U(1) param- eterized by k = 2 real coordinates 0 ≤ φ 2π, 0 ≤ θ ≤ π/2, and a diagonal unitary matrix Ud, that is Û = ÛdÛc, where Ûc = cos θ − sin θe−iφ sin θeiφ cos θ , ˆ Ud = e−iφ1 0 0 eiφ2 . (2.4) Then the differential dÛ of the matrix Û = ÛdÛc has the following form: ˆ dU = −[dθ sin θ + i cos θdφ1 ]e−iφ1 e−i(φ1 + φ) [−dθ cos θ + i(dφ1 + dφ) sin θ] ei(φ+ φ2 ) [dθ cos θ + i(dφ + dφ2 ) sin θ] [−dθ sin θ + idφ2 cos θ]eiφ2 , (2.5) which yields the length element and the induced Riemannian metric: (ds)2 = Tr dÛdÛ∗ (2.6) = 2(dθ)2 + (dφ1)2 + (dφ2)2 + 2 sin2 θ(dφ)2 +2 sin2 θ(dφ dφ1 + dφ dφ2). We see that the nonzero entries of the Riemannian metric tensor gmn in this case are g11 = 2, g22 = g33 = 1, g44 = 2 sin2 θ, g24 = g42 = g34 = g43 = sin2 θ, so that the determinant det [gmn] = 4 sin2 θ cos2 θ. Finally, the induced integration measure on the group U(2) is given by dµ(Û) = 2 sin θ cos θ dθ dφ dφ1 dφ2. (2.7) It is immediately clear that the above expression is invariant, by construction, with respect to multiplications Û → V̂ Û, for any fixed unitary matrix V from the same group. Therefore, Eq.(2.7) is just the Haar measure on the group. We will make use of these ideas several times in our lectures. Let us now concentrate on the N2 −dimensional subspace of Hermitian matrices in the 2N2 − dimensional space of all complex matrices of a given size N. The Her- miticity condition Ĥ = Ĥ∗ ≡ ĤT amounts to imposing the following restric- tions on the coordinates: xij = xji, yij = −yji. Such a restriction from the space of general complex matrices results in the length and volume element on the subspace of Hermitian matrices: (ds)2 = Tr dĤdĤ∗ = i (dxii)2 + 2 ij (dxij)2 + (dyij)2 (2.8) dµ(Ĥ) = 2 N (N −1) 2 i dxii ij dxijdyij. (2.9)
- 49. Introduction to the random matrix theory: Gaussian Unitary Ensemble and beyond 35 Obviously, the length element (ds)2 = TrdĤdĤ∗ is invariant with respect to an automorphism (a mapping of the space of Hermitian matrices to itself) by a similarity transformation Ĥ → U−1 ĤÛ, where Û ∈ U(N) is any given unitary N × N matrix: Û∗ = Û−1 . Therefore the corresponding integration measure dµ(Ĥ) is also invariant with respect to all such “rotations of the basis”. The above-given measure dµ(Ĥ) written in the coordinates xii, xij, yij is frequently referred to as the “flat measure”. Let us discuss now another, very important coordinate system in the space of Hermitian matrices which will be in the heart of all subsequent discussions. As is well-known, every Hermitian matrix Ĥ can be represented as Ĥ = ÛΛ̂Û−1 , Λ̂ = diag(λ1, . . . , λN ), Û∗ Û = ˆ I, (2.10) where real −∞ λk ∞, k = 1, . . . , N are eigenvalues of the Hermitian matrix, and rows of the unitary matrix Û are corresponding eigenvectors. Generically, we can consider all eigenvalues to be simple (non-degenerate). More precisely, the set of matrices Ĥ with non-degenerate eigenvalues is dense and open in the N2 -dimensional space of all Hermitian matrices, and has full measure (see [3], p.94 for a formal proof). The correspondence Ĥ → Û ∈ U(N), Λ̂ is, however, not one-to-one, namely Û1Λ̂Û−1 1 = Û2Λ̂Û−1 2 if Û−1 1 Û2 = diag eiφ1 , . . . , eiφN for any choice of the phases φ1, . . . , φN . To make the correspondence one-to-one we therefore have to restrict the unitary matrices to the coset space U(N)/U(1) ⊗ . . . ⊗ U(1), and also to order the eigenvalues, e.g. requiring λ1 λ2 . . . λN . Our next task is to write the integration measure dµ(Ĥ) in terms of eigenvalues Λ̂ and matrices Û. To this end, we differentiate the spectral decomposition Ĥ = ÛΛ̂Û∗ , and further exploit: d Û∗ Û = dÛ∗ Û + Û∗ dÛ = 0. This leads to dĤ = Û dΛ̂ + Û∗ dÛΛ̂ − Λ̂Û∗ dÛ Û∗ . (2.11) Substituting this expression into the length element (ds)2 , see Eq.(2.8), and us- ing the short-hand notation δÛ for the matrix Û∗ dÛ satisfying anti-Hermiticity condition δÛ∗ = −δÛ, we arrive at: (ds)2 = Tr dΛ̂ 2 + 2dΛ̂ δÛΛ̂ − Λ̂δÛ (2.12) + δÛΛ̂ 2 + Λ̂δÛ 2 − 2δÛΛ̂2 δÛ . Taking into account that Λ̂ is purely diagonal, and therefore the diagonal entries of the commutator δÛΛ̂ − Λ̂δÛ are zero, we see that the second term in Eq.(2.12) vanishes. On the other hand, the third and subsequent
- 50. 36 Yan V. Fyodorov terms when added up are equal to 2Tr δÛΛ̂δÛΛ̂ − δÛ2 Λ̂2 = 2 ij δUijλjδUjiλi − λ2 i δUijδUji = − ij (λi − λj)2 δUjiδUij which together with the first term yields the final expression for the length element in the “spectral” coordinates (ds)2 = i (dλi)2 + ij (λi − λj)2 δUijδUij (2.13) where we exploited the anti-Hermiticity condition −δUji = δUij. Introducing the real and imaginary parts δUij = δpij +iδqij as independent coordinates we can calculate the corresponding integration measure dµ(Ĥ) according to the rule in Eq.(2.3), to see that it is given by dµ(Ĥ) = ij (λi − λj)2 i dλi × dM(Û) . (2.14) The last factor dM(U) stands for the part of the measure depending only on the U−variables. A more detailed consideration shows that, in fact, dM(Û) ≡ dµ(Û), which means that it is given (up to a constant factor) by the invariant Haar measure on the unitary group U(N). This fact is however of secondary importance for the goals of the present lecture. Having an integration measure at our disposal, we can introduce a probabil- ity density function (p.d.f.) P(Ĥ) on the space of Hermitian matrices, so that P(Ĥ)dµ(Ĥ) is the probability that a matrix Ĥ belongs to the volume element dµ(Ĥ). Then it seems natural to require for such a probability to be invariant with respect to all the above automorphisms, i.e. P(Ĥ) = P Û∗ ĤÛ . It is easy to understand that this “postulate of invariance” results in P being a function of N first traces TrĤn , n = 1, . . . , N (the knowledge of first N traces fixes the coefficients of the characteristic polynomial of Ĥ uniquely, and hence the eigenvalues. Therefore traces of higher order can always be expressed in terms of the lower ones). Of particular interest is the case P(Ĥ) = C exp −Tr Q(Ĥ), Q(x) = a2jx2j + . . . + a0, (2.15) where 2j ≤ N, the parameters a2l and C are real constants, and a2j 0. Observe that if we take Q(x) = ax2 + bx + c, (2.16) then e−T r Q(Ĥ) takes the form of the product e−a[ i x2 ii +2 i j (x2 ij +y2 ij )]e−b i xii e−cN (2.17) = e−cN N i=1 e−ax2 ii −bxii ij e−2ax2 ij ij e−2ay2 ij .
- 51. Introduction to the random matrix theory: Gaussian Unitary Ensemble and beyond 37 We therefore see that the probability distribution of the matrix Ĥ can be rep- resented as a product of factors, each factor being a suitable Gaussian distribu- tion depending only on one variable in the set of all coordinates xii, xij, yij. Since the same factorization is valid also for the integration measure dµ(Ĥ), see Eq.(2.9), we conclude that all these N2 variables are statistically independent and Gaussian-distributed. A much less obvious statement is that if we impose simultaneously two requirements: • The probability density function P(Ĥ) is invariant with respect to all conjugations Ĥ → Ĥ = U−1 ĤÛ by unitary matrices Û, that is P(Ĥ ) = P(Ĥ); and • the N2 variables xii, xij, yij are statistically independent, i.e. P(Ĥ) = N i=1 fi(xii) N ij f (1) ij (xij)f (2) ij (yij), (2.18) then the function P(Ĥ) is necessarily of the form P(Ĥ) = Ce−(aTrĤ2 +bTrĤ+cN), for some constants a 0, b, c. The proof for any N can be found in [2], and here we just illustrate its main ideas for the simplest, yet nontrivial case N = 2. We require invariance of the distribution with respect to the conjugation of Ĥ by Û ∈ U(2), and first consider a particular choice of the unitary matrix Û = 1 −θ θ 1 corresponding to φ = φ1 = φ2 = 0, and small values θ 1 in Eq.(2.4). In this approximation the condition Ĥ = U−1 ĤÛ amounts to x 11 x 12 + iy 12 x 12 − iy 12 x 22 (2.19) = x11 + 2θx12 x12 + iy12 + θ (x22 − x11) x12 − iy12 + θ (x22 − x11) x22 − 2θx12 , where we kept only terms linear in θ. With the same precision we expand the factors in Eq.(2.18): f1(x 1) = f1(x1) 1 + 2θx12 1 f1 df1 dx11 , f2(x 22) = f2(x22) 1 − 2θx12 1 f2 df2 dx22 f (1) 12 (x 12) = f (1) 12 (x12) 1 + θ(x22 − x11) 1 f (1) 12 df (1) 12 dx12 , f (2) 12 (y 21) = f (2) 12 (y12). The requirements of statistical independence and invariance amount to the product of the left-hand sides of the above expressions to be equal to the product of the right-hand sides, for any θ. This is possible only if: 2x12 d ln f1 dx11 − d ln f2 dx22 + (x22 − x11) d ln f (1) 12 dx12 = 0, (2.20)
- 52. 38 Yan V. Fyodorov which can be further rewritten as 1 (x22 − x11) d ln f1 dx11 − d ln f2 dx22 = const = 1 2x12 d ln f (1) 12 dx12 , (2.21) where we used that the two sides in the equation above depend on essentially different sets of variables. Denoting const1 = −2a, we see immediately that f (1) 12 (x12) ∝ e−2ax2 12 , and further notice that d ln f1 dx11 + 2ax11 = const2 = d ln f2 dx22 + 2ax22 by the same reasoning. Denoting const2 = −b, we find: f1(x11) ∝ e−ax2 11 −bx11 , f2(x22) ∝ e−ax2 22 −bx22 , (2.22) and thus we are able to reproduce the first two factors in Eq.(2.17). To repro- duce the remaining factors we consider the conjugation by the unitary matrix ˆ Ud = 1 − iα 0 0 1 + iα , which corresponds to the choice θ = 0, φ1 = φ2 = −α = in Eq.(2.4), and again we keep only terms linear in the small parameter α 1. Within such a precision the transformation leaves the diagonal entries x11, x22 unchanged, whereas the real and imaginary parts of the off-diagonal entries are transformed as x 12 = x12 − 2αy12, y 12 = y12 + 2αx12. In this case the invariance of the p.d.f. P(Ĥ) together with the statistical independence of the entries amount, after straightforward manipulations, to the condition 1 x12 d ln f (1) 12 dx12 = 1 y12 d ln f (2) 12 dy12 which together with the previously found f (1) 12 (x12) yields f (2) 12 (y12) ∝ e−2ay2 12 , completing the proof of Eq.(2.17). The Gaussian form of the probability density function, Eq.(2.17), can also be found as a result of rather different lines of thought. For example, one may invoke an information theory approach a la Shanon-Khinchin and define the amount of information I[P(Ĥ)] associated with any probability density function P(Ĥ) by I[P(Ĥ)] = − dµ(Ĥ) P(Ĥ) ln P(Ĥ) (2.23)
- 53. Introduction to the random matrix theory: Gaussian Unitary Ensemble and beyond 39 This is a natural extension of the corresponding definition I[p1, . . . , pm] = − m l=1 pm ln pm for discrete events 1, ..., m. Now one can argue that in order to have matrices Ĥ as random as possible one has to find the p.d.f. minimizing the information associated with it for a certain class of P(H) satisfying some conditions. The conditions usually have a form of constraints ensuring that the probability density function has desirable properties. Let us, for example, impose the only requirement that the ensemble average for the two lowest traces TrĤ, TrĤ2 must be equal to certain prescribed values, say E TrĤ = b and E TrĤ2 = a 0, where the E [. . .] stand for the expectation value with respect to the p.d.f. P(H). Incorporating these constraints into the minimization procedure in a form of Lagrange multipliers ν1, ν2, we seek to minimize the functional I[P(Ĥ)] = − dµ(Ĥ) P(Ĥ) ln P(Ĥ) − ν1TrĤ − ν2TrĤ2 . (2.24) The variation of such a functional with respect to δP(H) results in δI[P(Ĥ)] = − dµ(Ĥ) δP(Ĥ) 1 + ln P(Ĥ) − ν1TrĤ − ν2TrĤ2 = 0 (2.25) possible only if P(Ĥ) ∝ exp{ν1TrĤ + ν2TrĤ2 } again giving the Gaussian form of the p.d.f. The values of the Lagrange mul- tipliers are then uniquely fixed by constants a, b, and the normalization con- dition on the probability density function. For more detailed discussion, and for further reference see [2], p.68. Finally, let us discuss yet another construction allowing one to arrive at the Gaussian Ensembles exploiting the idea of Brownian motion. To start with, consider a system whose state at time t is described by one real vari- able x, evolving in time according to the simplest linear differential equation d dt x = −x describing a simple exponential relaxation x(t) = x0e−t towards the stable equilibrium x = 0. Suppose now that the system is subject to a random additive Gaussian white noise ξ(t) function of intensity D 1 , so that the corresponding equation acquires the form d dt x = −x + ξ(t), Eξ [ξ(t1)ξ(t1)] = Dδ(t1 − t2), (2.26) 1 The following informal but instructive definition of the white noise process may be helpful for those not very familiar with theory of stochastic processes. For any 0 t 2π and integer k ≥ 1 define the random function ξk (t) = 2/π k n=0 an cos nt, where real coefficients an are all independent, Gaussian distributed with zero mean E[an ] = 0 and variances E[a2 0] = D/2 and E[a2 n ] = D for 1 ≤ n ≤ k. Then one can, in a certain sense, consider white noise as the limit of ξk (t) for k → ∞. In particular, the Dirac δ(t − t ) is approximated by the limiting value of sin [(k+1/2)(t−t )] 2π sin (t−t)/2
- 54. 40 Yan V. Fyodorov where Eξ[. . .] stands for the expectation value with respect to the random noise. The main characteristic property of a Gaussian white noise process is the following identity: Eξ exp b a ξ(t)v(t)dt = exp D 2 b a v2 (t)dt (2.27) valid for any (smooth enough) test function v(t). This is just a direct general- ization of the standard Gaussian integral identity: ∞ −∞ dq √ 2πa e− 1 2a q2 +qb = e ab2 2 , (2.28) valid for Re a 0, and any (also complex) parameter b. For any given realization of the Gaussian random process ξ(t) the solution of the stochastic differential equation Eq.(2.26) is obviously given by x(t) = e−t x0 + t 0 eτ ξ(τ)dτ . (2.29) This is a random function, and our main goal is to find the probability density function P(t, x) for the variable x(t) to take value x at any given moment in time t, if we know surely that x(0) = x0. This p.d.f. can be easily found from the characteristic function F(t, q) = Eξ e−iqx(t) = exp −iqx0e−t − Dq2 4 (1 − e−2t ) (2.30) obtained by using Eqs. (2.27) and (2.29). The p.d.f. is immediately recovered by employing the inverse Fourier transform: P(t, x) = ∞ −∞ dq 2π eiqx Eξ e−iqx(t) (2.31) = 1 πD(1 − e−2t) exp − (x − x0e−t ) 2 D(1 − e−2t) . The formula Eq.(2.31) is called the Ornstein-Uhlenbeck (OU) probability density function, and the function x(t) satisfying the equation Eq.(2.26) is known as the O-U process. In fact, such a process describes an interplay between the random “kicks” forcing the system to perform a kind of Brownian motion and the relaxation towards x = 0. It is easy to see that when time grows the OU p.d.f. “forgets” about the initial state and tends to a stationary (i.e. time-independent) universal Gaussian distribution: P(t → ∞, x) = 1 √ πD exp − x2 D . (2.32)
- 55. Other documents randomly have different content
- 56. made no attempt to modify for these the provisions of the General Consolidated Order of 1847, the effect of which upon the workhouse administration of the period we have already described.[358] Meanwhile the workhouse schools continued to improve very slowly in educational efficiency. The policy of the Central Authority was apparently to develop industrial training—agricultural work, the simpler handicrafts, and domestic service—on the model of the Quatt School in Shropshire. Whether or not this industrial work militated against more intellectual accomplishments is a moot point, but we hear of the reports of 'the stagnant dulness of workhouse education' which annually proceed from Her Majesty's Inspectors of Schools.[359] Whether or not from a certain divergence of aim between the departments, the connection was in 1863 severed,[360] and the Poor Law Board thenceforward had its own inspectors of Poor Law Schools, whose criticisms and complaints, all in favour of the large district schools as compared with the single union school, appear from 1867 onward in the Annual Reports.[361] At the very end of the period we may note the beginning of a reaction against the barrack schools. It was pointed out by those acquainted with the Scottish system of boarding-out, as well as by persons experienced in English Poor Law administration, that these expensive boarding schools were not answering so well as their admirers claimed, especially as regards the girls. During 1866-9 the alternative of boarding-out children in private families at 4s. a week (now 5s.) was warmly discussed, and experimentally adopted in a few places.[362] In 1869 the Central Authority so far yielded to the criticisms made upon these institutions as to permit, under elaborate restrictions and safeguards, the boarding-out, in families beyond the limits of the union, of the comparatively small class of children who were actually or practically orphans.[363] In these cases all idea of making the condition of the pauper child less eligible than that of the lowest independent labourer was definitely abandoned.
- 57. The whole concern of the Central Authority was to see that the provision for the boarded-out child was good and complete. Far from being assimilated to the children of the lowest independent labourers, the boarded-out children were only to be entrusted to specially selected families superior to the lowest, who undertook to bring them up as their own, to provide proper food, clothing and washing, to train them in good habits as well as in suitable domestic and industrial work, and to make them regularly attend school and place of worship. For all this the foster parents were to receive with each child a sum three or four times as great as was, with the sanction of the Central Authority, commonly allowed for the maintenance of each of the couple of hundred thousand children at that date on outdoor relief; and which (as Professor Fawcett vainly objected) was far in excess of what the ordinary labourer could afford to expend on his own children.[364] A plan, observed Mr. Fowle, which cannot be defended on any sound principles of Poor Law.[365] It is indeed impossible, says Mr. Mackay in this connection, to deny that apparently every provision for pauper children may be regarded as a contravention of this rule.... Professor Fawcett's ... argument has been tacitly neglected.[366] E.—The Sick We have shown that, between 1834 and 1847, it was not contemplated that persons actually sick would be received in the workhouse, and that there was no trace in the documents of any desire on the part of the Central Authority to interfere with the usual practice of granting to them outdoor relief, which had not been in any way condemned or discredited by the 1834 Report. The same may be said of the Statutes, Orders, and Circulars of 1847-71. We find no suggestion that the boards of guardians ought not to grant outdoor relief in cases of sickness, or that sick paupers ought to be relieved in the workhouse. On the contrary, the exceptions specifically made in favour of sick persons seem to be even widened in scope. Thus, in 1848, the Central Authority laid it down that widows with illegitimate children were not to be refused outdoor
- 58. relief, if the children were sick.[367] By the Outdoor Relief Regulation Order of December 1852, it was definitely provided that outdoor relief might be given in case of sickness in the family, even if the head of the family was simultaneously earning wages.[368] The same policy was embodied in the corresponding General Order issued on 1st January 1869, to certain Metropolitan unions.[369] Further, in the panic about cholera in 1866, the Central Authority informed the boards of guardians by circular that in cases of emergency they might call in any medical and other assistance that was needed, and even provide whatever sustenance, clothing, etc., was required,[370] apparently irrespective of destitution and of all General Orders, etc., to the contrary. Moreover, early in this period we note the beginning of the special definition of destitution as regards medical relief which has since been acted upon, that is to say, the inability to pay for the medical attendance that the nature of the case requires. Thus it was declared by the Central Authority in 1848 that the parish doctor might attend sick servants living in their master's household, who were plainly not destitute in the ordinary sense, as not being without food and lodging, but who, if there were no wages due to them, might be unable to pay for medical attendance.[371] A similar line of thought may be traced in that provision of the Act of 1851 which authorised boards of guardians to make annual subscriptions out of the poor rate to public hospitals and infirmaries, to enable these non-pauper institutions the better to provide for the poor.[372] The sick wards of the workhouses, as the Central Authority explained in 1869, were originally provided for the cases of paupers in the workhouse who might be attacked by illness; and not as State hospitals into which all the sick poor of the country might be received for medical treatment and care. So far is this, indeed, from being the case that at least two-thirds of the sick poor receive medical attendance and treatment in their own homes.[373] When in 1869-71, the Central Authority obtained elaborate reports showing, for all parts of England, the practice that prevailed of normally giving outdoor relief to the sick, and of taking them into the workhouse infirmaries only when this was called for by (a) the nature of the
- 59. disease, (b) the wishes of the patient, or (c) the nature of the home, and then only where suitable infirmary accommodation was available, there is no indication that any objection was entertained to the policy of outdoor relief to this large class.[374] What is new in this period is the appearance, as a positive policy, of bringing pressure to bear on the boards of guardians to improve the quality of the medical attendance and medicine supplied. This led to an explicit disavowal, so far as regards the sick paupers, of any application to them of the principle of making the pauper's condition less eligible than that of the lowest grade of independent labourers. It is noteworthy that this new departure applied to outdoor medical relief quite as much as to institutional medical treatment, in which it has subsequently been sometimes excused on the ground that the superior treatment is accompanied by a loss of liberty. The new departure took three directions. It was definitely laid down that the medical attendance afforded to the outdoor paupers was to be of good quality, and thus necessarily above that obtained by the poorest independent labourer, or even by the poor generally. This was the outcome of a long campaign on behalf of the poorer members of the medical profession, of which Wakley was the leader in the House of Commons, and the Lancet the efficient organ.[375] In 1853 the Poor Law Board considered that the qualifications of the Poor Law medical officers ought to be such as to ensure for the poor a degree of skill in their medical attendants equal to that which can be commanded by the more fortunate classes of the community.[376] On the suggestion of the House of Commons Committee on Poor Relief[377] it was authoritatively enjoined on boards of guardians in 1865 by a special circular that they were to supply freely quinine, cod-liver oil, and other expensive medicines to the sick poor;[378] although it must have been plain that such things were beyond the reach of the independent labourers consulting the sixpenny doctor, and even beyond the usual resources of the provident dispensaries of the period.[379] Finally, in 1867, the Metropolitan Poor Act authorised the establishment throughout London of Poor Law dispensaries. These institutions were
- 60. consistently pressed on the Metropolitan boards of guardians by the Central Authority, as having been successful in Ireland in reducing the amount of sickness among the poor, and as ensuring, not only regular and more successful medical attention, but also a sufficient supply of medicines and medical appliances of standard quality.[380] By this elaborate systematisation of outdoor medical relief, the Central Authority not only put within the reach of the sick paupers medical attendance far superior to that accessible to the lowest grade of independent labourers, but even placed the sick pauper in the Metropolis, without loss of liberty, in a position equal to that of the superior artisan subscribing to a good provident dispensary. The most remarkable change of front was, however, that relating to the institutional treatment of the sick. Down to 1847, it is not too much to say that what may be called the hospital branch of Poor Law administration[381] was ignored alike by Parliament, public opinion, and the Central Authority. We have shown that the institutional provision for the sick was not so much as mentioned in the Report of 1834, and that it remained practically ignored in all the Orders, Circulars, and Reports of the Poor Law Commissioners. The same is true of the first eighteen years of the Poor Law Board. Few and far between are the incidental references to the sick wards of the workhouses. There is not even a hint of a suggestion that relief to the sick poor could most advantageously take the form of an offer of the House. On the contrary, it was held in 1848 that applicants for admission suffering from fever might even be refused admission, the relieving officer being enjoined to find lodging elsewhere for them,[382] though how this was to be done the Central Authority did not, in 1848, say. In 1857, the Metropolitan Boards of Guardians were recommended to send such cases to the London Fever Hospital[383] (involving a payment by the guardians of 7s. weekly). Finally, in 1864-5, we have an outburst of public indignation, at the condition into which the sick wards of the workhouses had been allowed to drift. The death of a pauper in Holborn workhouse, and of another in St. Giles's workhouse, under conditions which seemed to point to inhumanity and neglect, led to
- 61. an enquiry by three doctors (Anstie, Carr, and Ernest Hart), commissioned by the Lancet newspaper, the formation of an Association for improving the condition of the sick poor, and a deputation to the Poor Law Board.[384] The publication of various reports on the workhouse infirmaries, in which terrible deficiencies were revealed,[385] led to public discussion and Parliamentary debates. The Central Authority at once accepted the new standpoint. It made no attempt to resist the provision of the necessarily costly institutional treatment for the sick poor, whether or not their ailments were infectious or otherwise dangerous to the public. The progressive improvement of the hospital branch of Poor Law administration, to use the phrase of the Central Authority itself, which had in the preceding thirty years grown up unawares, was now definitely accepted as an important feature of its policy. Statutory powers were obtained for the provision of hospitals in the Metropolis by combinations of boards of guardians. Urgent letters were written pressing the boards of guardians to embark on the expenditure required to enable them to provide efficiently for the sick paupers.[386] From 1865 onward, we see the Central Authority, on the public-spirited initiative of Mr. W. Rathbone and the Liverpool Select Vestry, pressing on the boards of guardians the employment of salaried and qualified nurses to attend to the sick paupers, whatever their complaints.[387] We have even in 1867, so far as the sick are concerned, the explicit disavowal by the Central Authority of the very idea of the deterrent workhouse, which had formed so prominent a part of the policy of 1834-1847. Mr. Gathorne Hardy, speaking as President of the Poor Law Board, said there is one thing ... which we must peremptorily insist on, namely, the treatment of the sick in the infirmaries being conducted on an entirely separate system, because the evils complained of have mainly arisen from the workhouse management—which must to a great degree be of a deterrent character—having been applied to the sick, who are not proper objects for such a system.[388] At first the new policy of the Central Authority for the institutional treatment of the sick took the form of the erection of special
- 62. hospitals by Sick Asylum Districts.[389] Presently, however, it came to the conclusion that this involved an unnecessary expense, and that it would be cheaper to revert to the idea of the Report of 1834, and use the existing workhouse buildings by a system of classification by institutions.[390] So definitely was this recognised as a reversion to 1834 that the Central Authority actually quoted the passage of the 1834 Report in justification of its plan.[391] From this point may be dated the adoption of the policy of the provision, in connection with the workhouse, but practically as a separate institution, of what is now called the Poor Law Infirmary.[392] In 1870 the Central Authority took pains to collect special statistics as to the extent to which this recently developed provision for the sick was being taken advantage of. It observes (and, significantly enough, without expression of disapproval) that the numbers on the lists of relieving officers may be swollen by poor persons who in previous years, though really poor, refrained from coming on the rates, but whom changes in the law or in the mode of its administration have since attracted.[393] Workhouses, it notes, originally designed mainly as a test for the able-bodied, have, especially in the large towns, been of necessity gradually transformed in to infirmaries for the sick. The higher standard for hospital accommodation has had a material effect upon the expenditure. So again it has been considered necessary to attach to workhouses separate fever wards; and wherever it was possible, these wards have been isolated by the erection of a separate building.[394] The extent to which the Poor Law had become the public doctor was indeed remarkable. The number of persons on outdoor relief who were actually sick, apart from mere old age infirmity, and without their families, was found to be 13 per cent of the whole, equal to about 119,000. The number in the workhouses who were actually sick, irrespective of the vast number of old people disabled by old age, but not actually upon the sick list, varied in different unions from 14 to 39 per cent in the provinces, and up to nearly 50 per cent in some Metropolitan Unions; amounting, for the whole country, to about 60,000 actual sick-bed cases.[395] Taking indoor and outdoor patients together, the
- 63. total simultaneously under medical treatment in the twelfth week of the half-year ending Lady Day 1870, was estimated at 173,000, being three quarters of one per cent of the population, and perhaps one out of four of all the persons under medical treatment in the whole population. The story from this date is one continuous record, on the one hand of an ever-increasing number of patients treated, and, on the other, of never slackening pressure by the Central Authority to induce apathetic or parsimonious boards of guardians to expend money in making both the outdoor medical service and the workhouse infirmaries as efficient and as well adapted and as well equipped for the alleviation and cure of their patients—without the least notion of the principle of less eligibility—as the most scientifically efficient hospitals and State medical service in any part of the world. After 1867, indeed, there was developed, for the Metropolitan paupers suffering from infectious diseases, the splendid hospital system of the Metropolitan Asylums Board.[396] At the very end of the existence of the Poor Law Board, Mr. Goschen seems almost to have been contemplating a yet further extension. The economical and social advantages, he observed, of free medicine to the poorer classes generally as distinguished from actual paupers, and perfect accessibility to medical advice at all times under thorough organisation, may be considered as so important in themselves as to render it necessary to weigh with the greatest care all the reasons which may be adduced in their favour.[397] F.—Persons of Unsound Mind It is difficult to discover what was the policy of the Central Authority during this period with regard to lunatics, idiots, and the mentally defective. Lunacy had always been, and remained, a ground of exception from the prohibition to grant outdoor relief. The provision of a lodging for a lunatic was, moreover, an exception to the prohibition of the payment of rent for a pauper. As a result of these exceptions, there were on 1st January 1852, 4107 lunatics and idiots on outdoor relief,[398] and this number had increased by 1859 to 4892[399] and by 1870 to 6199.[400] The Central Authority took no
- 64. steps to require or persuade boards of guardians not to grant outdoor relief to lunatics, nor yet to get any appropriate provision made for them in the great general workhouses on which it had insisted. Parliament in 1862 (in order to relieve the pressure on lunatic asylums) expressly authorised arrangements to be made for chronic lunatics to be permanently maintained in workhouses, under elaborate provisions for their proper care.[401] These arrangements would have amounted, in fact, to the creation, within the workhouse, of wards which were to be in every respect as well equipped, as highly staffed, and as liberally supplied as a regular lunatic asylum.[402] The Central Authority transmitted the Act to the boards of guardians, observing, with what almost seems like sarcasm, that it was not aware of any workhouse in which any such arrangements could conveniently be made;[403] and the provisions of this Act were, we believe, never acted upon. Whilst consistently objecting to the retention in workhouses of lunatics who were dangerous, or who were deemed curable, we do not find that the Central Authority ever insisted on there being a proper lunatic ward for the persons of unsound mind who were necessarily received, for a longer or shorter period, in every workhouse.[404] Moreover, the Central Authority took no steps to get such persons removed to lunatic asylums. In 1845 it had agreed with the Manchester Board of Guardians (who did not want to make any more use of the county asylum than they could help) that they were justified in retaining in the workhouse any lunatics whom their own medical officer did not consider proper to be confined in a lunatic asylum.[405] In 1849 it expressly laid it down that a weak-minded pauper or, as we now say, a mentally defective, must either be a lunatic, and be certified and treated as such, or not a lunatic, in which case no special treatment could be provided for him or her in the one general workhouse to which the Central Authority still adhered.[406] We can find no indication of policy as to whether it was recommended that such mentally defectives should be granted outdoor relief, or (as one can scarcely believe) required to inhabit a workhouse which made no provision for them.[407]
- 65. The explanation of this paralysis of the Central Authority, as regards the policy to be pursued with persons of unsound mind, is to be found, we believe, in the existence and growth during this period of the rival authority of the Lunacy Commissioners, who had authority over all persons of unsound mind, whether paupers or not. The Lunacy Commissioners had not habitually in their minds the principle of less eligibility; and they were already, between 1848 and 1871, making requirements with regard to the accommodation and treatment of pauper lunatics that the Poor Law authorities regarded as preposterously extravagant. The records of the boards of guardians show visits of the inspectors of the Lunacy Commissioners, and their perpetual complaints of the presence of lunatics and idiots in the workhouses without proper accommodation; mixed up with the sane inmates to the great discomfort of both;[408] living in rooms which the Lunacy Commissioners considered too low and unventilated, with yards too small and depressing, amid too much confusion and disorder, for the section of the paupers for whom they were responsible.[409] Such reports, officially communicated to the Poor Law Board, seem to have been merely forwarded for the consideration of the board of guardians concerned. But other action was not altogether wanting. Under pressure from the Lunacy Commissioners, the Central Authority asked, in 1857, for more care in the conveyance of lunatics;[410] urged, in 1863, a more liberal dietary for lunatics in workhouses;[411] in 1867 it reminded the boards of guardians that lunatics required much food, especially milk and meat;[412] it was thought very desirable that the insane inmates ... should have the opportunity of taking exercise;[413] it concurred with the Visiting Commissioner in deeming it desirable that a competent paid nurse should be appointed for the lunatic ward, in a certain workhouse; [414] it suggested the provision of leaning chairs in another workhouse;[415] and, in yet another, the desirability of not excluding the persons of unsound mind from religious services.[416] In 1870 it issued a circular, transmitting the rules made by the Lunacy Commissioners as to the method of bathing lunatics, for the careful
- 66. consideration of the boards of guardians.[417] But we do not find that the Central Authority issued any Order amending the General Consolidated Order of 1847, which, it will be remembered, did not include among its categories for classification either lunatics, idiots, or the mentally defective; and the Central Authority did not require any special provision to be made for them. The policy of the Lunacy Commissioners was to get provision made in every county for all the persons of unsound mind, whatever their means, in specially organised lunatic asylums in which the best possible arrangements should be made for their treatment and cure irrespective of cost, and altogether regardless of making the condition of the pauper lunatic less eligible than that of the poorest independent labourer. Unlike the provision for education, and that for infectious disease, the cost of this national (and as we may say communistic) provision for lunatics was a charge upon the poor rate. Under the older statutes, the expense of maintaining the inmates of the county lunatic asylums was charged to the Poor Law authorities of the parishes in which they were respectively settled; and the boards of guardians were entitled to recover it, or part of it, from any relatives liable to maintain such paupers, even in cases in which the removal to the asylum was compulsory and insisted on in the public interest.[418] The great cost to the poor rate of lunatics sent to the county lunatic asylums, and the difficulty of recovering the amount from their relatives, prevented the whole-hearted adoption, either by the boards of guardians, or the Central Authority, of the policy of insisting on the removal of persons of unsound mind to the county asylums. For the imbeciles and idiots of the Metropolitan Unions, provision was made after 1867 in the asylums of the Metropolitan Asylums Board.[419] But no analogous provision for those of other unions was made. The result was that, amid a great increase of pauper lunacy, the proportion of the paupers of unsound mind who were in lunatic asylums did not increase.[420] On the other hand the indisposition of the Central Authority to so amend the General Consolidated Order of 1847 as to put lunatics in a separate category, and require suitable accommodation and treatment for
- 67. them—an indisposition perhaps strengthened by the very high requirements on which the Lunacy Commissioners would have insisted—stood in the way of any candid recognition of the fact that for thousands of lunatics, idiots, and mentally defectives, the workhouse had, without suitable provision for them, and often to the unspeakable discomfort of the other inmates, become a permanent home. G.—Defectives During this period, the blind, the deaf and dumb, and the lame and deformed were increasingly recognised by Parliament as classes for whom the Poor Law authorities might, if they chose, provide expensive treatment. This was done by authorising boards of guardians, if they chose, to pay for their maintenance, whether children or adults, in special institutions.[421] We do not find that the Central Authority suggested the adoption of this or any other policy or gave any lead to the boards of guardians with regard to these cases.[422] H.—The Aged and Infirm We have shown that neither the Report of 1834 nor the Central Authority between 1834 and 1847 even suggested any departure from the common practice of granting outdoor relief to the aged and infirm. This continued, so far as the official documents show, to be the policy of the Central Authority during the whole of the period 1847-1871.[423] The only two references to the subject in the Orders and Circulars of this period assume that the aged and infirm will normally be relieved in their own homes. Thus, in 1852, in commenting on the provision requiring the weekly payment of relief, the Central Authority said, as to the cases in which the pauper is too infirm to come every week for the relief, it is on many accounts advantageous that the relieving officer should, as far as possible, himself visit the pauper, and give the relief at least weekly.[424] And in the first edition of the Out-relief Regulation Order of 1852 (that of 25th August 1852) the Central Authority, far from prohibiting outdoor
- 68. relief to persons indigent and helpless from age, sickness, accident, or bodily or mental infirmity, formally sanctioned this practice, by ordering that one third at least of such relief should be given in kind (viz., in articles of food or fuel, or in other articles of absolute necessity),[425] the object being expressly explained to be, not, as might nowadays have been imagined, the discouragement of such relief, but the prevention of its misappropriation.[426] This provision was objected to by boards of guardians up and down the country, on the ground that it would be a hardship to the aged and infirm poor. The Poplar Board of Guardians, for instance, stated that there are a large number of persons under the denomination of aged and infirm whom the guardians have, in their long practical experience, found it expedient and not objectionable to relieve wholly in money, feeling assured that it would be beneficially expended for their use, and that in consequence of their infirmity the relieving officer or his assistant, if necessary, is thereby enabled to conveniently relieve them at their own house.[427] The Norwich Guardians stated that it would be difficult to determine (especially for the aged and sick poor) what kind of food or articles should be given. They also communicated with forty other unions, summoning them to concerted resistance. [428] A deputation from most of the large and populous unions in the north of England ... and from several Metropolitan parishes, representing in the aggregate upwards of 2,000,000 of population, [429] assembled in London, and objected to nearly all the provisions of the Order. Accompanied by about twenty-five members of Parliament, the deputation waited on the Poor Law Board, and specially urged their objection to being compelled to give a third of all outdoor relief in kind. After two hours' argumentative discussion, Sir John Trollope said that the board would reconsider the whole Order, which need not in the meantime be acted upon; and he hinted at a probable modification of the Article relating to relief in kind.[430] In response to these objections, the Central Authority does not seem even to have suggested that outdoor relief to the aged and infirm was contrary to its principles. It first intimated its willingness to modify the Order if
- 69. its working proved to be accompanied with hardship to the aged or helpless poor[431] and then within a few weeks withdrew the provision altogether as regards any but the able-bodied.[432] It was expressly explained that the Order, as re-issued, was intended as a precaution against the injurious consequences of maintaining out of the poor rate able-bodied labourers and their families in a state of idleness, and that the Central Authority left to the boards of guardians full discretion as to the description of relief to be given to indigent poor of every other class.[433] From that date down to the abolition of the Poor Law Board in 1871, we can find in the documents no hint or suggestion that it disapproved of outdoor relief to the aged and infirm. On 1st January 1871, nearly half the outdoor relief was due to this cause.[434] I.—Non-Residents There was no change in the policy of preventing relief to paupers not resident within the union. The Outdoor Relief Regulation Order of 1852 embodied the prohibition with the same exceptions as had been contained in the Outdoor Relief Prohibitory Order of 1844, omitting, however, that of widows without children during the first six months of their widowhood. But, as has been already mentioned, at the very end of the period the Boarding-Out Orders of 1869, etc., permitted children to be maintained outside the union. J.—The Workhouse We have seen that between 1834 and 1847 the Central Authority turned directly away from the express recommendations of the 1834 Report with regard to the institutional accommodation of the paupers. Instead of a series of separate institutions appropriately organised and equipped for the several classes of the pauper population—the aged and infirm, the children, and the adult able- bodied—the Central Authority had got established, in nearly every union, one general workhouse; nearly everywhere the same cheap, homely building, with one common regimen, under one management, for all classes of paupers.
- 70. The justification for the policy which, as we have seen, Sir Francis Head induced the Central Authority to substitute for the recommendations of the 1834 Report, may have been his confident expectation, in 1835, that the use of the workhouse was only to serve as a test, which the applicants would not pass, and that there was accordingly no need to regard the workhouse building as a continuing home.[435] This was the view taken by Harriet Martineau, who, in her Poor Law Tales, describes the overseer of the depauperised parish as locking the door of the empty workhouse when it had completely fulfilled its purpose of a test by having made all the applicants prefer and contrive to be independent of poor relief. By 1847, however, it must have been clear that, even in the most strictly administered parishes, under the most rigid application of the Outdoor Relief Prohibitory Order, there would be permanently residing in the workhouse a motley crowd of the aged and infirm unable to live independently; the destitute chronic sick in like case; the orphans and foundlings; such afflicted persons as the village idiot, the senile imbecile, the deaf and dumb, and what we now call the mentally defective; together with a perpetually floating population of acutely sick persons of all ages; vagrants; girls with illegitimate babies; wives whose husbands had deserted them, or were in prison, in hospital, or in the Army or Navy; widows beyond the first months of their widowhood and other women unable to earn a livelihood; all sorts of ins and outs; and the children dragging at the skirts of all these classes. The workhouse population in 590 unions of England and Wales on 1st January 1849, was, in fact, 121,331.[436] The condition of these workhouse inmates, and the character of the regimen to which they were subjected, had been brought to public notice in 1847 in the notorious Andover case. The insanitary condition of the workhouses of the period as places of residence, and, in particular, their excessive death-rate, was repeatedly brought to notice not only by irresponsible agitators, but also by such competent statistical and medical critics as McCulloch and Wakley.[437] But the very idea of the general workhouse was now subjected to severe criticism. During the last ten years, said the
- 71. author of an able book in 1852, I have visited many prisons and lunatic asylums, not only in England, but in France and Germany. A single English workhouse contains more that justly calls for condemnation in the principle on which it is established than is found in the very worst prisons or public lunatic asylums that I have seen. The workhouse as now organised is a reproach and disgrace peculiar to England; nothing corresponding to it is found throughout the whole continent of Europe. In France the medical patients of our workhouses would be found in 'hopitaux'; the infirm aged poor would be in hospices; and the blind, the idiot, the lunatic, the bastard child and the vagrant would similarly be placed each in an appropriate but separate establishment. With us a common Malebolge is provided for them all; and in some parts of the country the confusion is worse confounded by the effect of Prohibitory Orders, which, enforcing the application of the notable workhouse- test, drive into the same common sink of so many kinds of vice and misfortune the poor man whose only crime is his poverty, and whose want of work alone makes him chargeable. Each of the buildings which we so absurdly call a workhouse is, in truth (1) a general hospital; (2) an almshouse; (3) a foundling house; (4) a lying-in hospital; (5) a school house; (6) a lunatic asylum; (7) an idiot house; (8) a blind asylum; (9) a deaf and dumb asylum; (10) a workhouse; but this part of the establishment is generally a lucus a non lucendo, omitting to find work even for able-bodied paupers. Such and so varied are the destinations of these common receptacles of sin and misfortune, of sorrow and suffering of the most different kinds, each tending to aggravate the others with which it is unnecessarily and injuriously brought into contact. It is at once equally shocking to every principle of reason and every feeling of humanity, that all these varied forms of wretchedness should be thus crowded together into one common abode, that no attempt should be made by law to classify them, and to provide appropriate places for the relief of each.[438] During the period now under review, 1847-71, we see the Central Authority becoming gradually alive to the draw-backs of this mixture
- 72. of classes. At first its remedy seems to have been to take particular classes out of the workhouse. We have already described the constant attempts, made from the very establishment of the Poor Law Board, to have the children removed to separate institutions and to get the vagrants segregated into distinct casual wards. It was the resistance and apathy of the boards of guardians that prevented these attempts being particularly successful,[439] and the Central Authority appears not to have felt able to issue peremptory orders on the subject. The policy of the Lunacy Commissioners drew many lunatics out of the workhouses, but this was more than made up by the increasing tendency to seclude the village idiot, so that the workhouse population of unsound mind actually increased. We do not find that there was during the whole period any alteration of the General Consolidated Order of 1847, upon which the regimen of the workhouse depended. In spite of the increasing number of the sick and the persons of unsound mind, the seven classes of workhouse inmates determined by that Order were adhered to, and received no addition, though the Poor Law Board favoured the sub- division of these classes so far as it was reasonably possible in the existing buildings, especially in the case of women. In a letter of 1854[440] it lamented the evil which arose from the association of girls, when removed from workhouse union schools, with women of bad character in the able-bodied women's ward, and wished that it could be prevented. At the same time it stated that in the smaller workhouses it was often impracticable to provide the accommodation which would be necessary in order to maintain a complete separation; and while pointing out that it was legally competent for the guardians (with its approval) to erect extra accommodation, by means of which this contamination could be avoided, the Central Authority did not even remotely suggest that it was the guardians' duty so to do. By 1860 it had given instructions that every new workhouse should be so constructed as to allow of the requisite classification.[441]
- 73. From about 1865 onwards we note a new spirit in all the circulars and letters relating to the workhouse. The public scandal caused by the Lancet inquiry into the conditions of the sick poor in the workhouses, and the official reports and Parliamentary discussions that ensued, seem to have enabled the Central Authority to take up a new attitude with regard both to workhouse construction and workhouse regimen. From this time forth the workhouse is recognised as being, not merely a test of destitution for the able- bodied, which they were not expected long to endure, but also the continuing home of large classes of helpless and not otherwise than innocent persons. Able-bodied people, reported the Medical Officer in 1867, are now scarcely at all found in them during the greater part of the year.... Those who enjoy the advantages of these institutions are almost solely such as may fittingly receive them, viz. the aged and infirm, the destitute sick and children. Workhouses are now asylums and infirmaries.[442] From now onwards we see the Central Authority always striving to improve the workhouse. In the Circulars of 1868 much attention was paid to the sufficiency of space and ventilation. It was required that parallel blocks of building should be so far apart as to allow free access to light and air; blocks connected at a right or acute angle were to be avoided. Ordinary wards were to be at least ten feet high and eighteen feet wide, the length depending on the number of inmates; 300 cubic feet of space were required for each healthy person in a dormitory, 500 for infirm persons able to leave the dormitory during the day, and 700 in a day and night room.[443] The Visiting Committee was to ascertain not merely whether the total number for which the workhouse is certified has been exceeded, but whether the number of any one class exceeds the accommodation available for it.[444] No wards were to be placed side by side without a corridor between them; the corridors were to be six feet wide, and ordinary dormitories were to have windows into them. Windows and fanlights into internal spaces were to be made to open to be used as
- 74. ventilators, and ventilation was also to be effected by special means, apart from the usual means of doors, windows, and fire- places, air-bricks being recommended as a simple method.[445] No rooms occupied by the inmates as sleeping-rooms were to be on the boundary of the workhouse site. Hot and cold water was to be distributed to the bath-rooms and sick wards. Airing yards for the inmates were to be of sufficient size—with a rider that if partially or wholly paved with stone or brick or asphalted or gas-tarred they are often better than if covered with gravel.[446] Yards for the children, sick, and aged were to be enclosed with dwarf walls and palisades where practicable, presumably with the object of giving a look-out, and making the yard slightly less prison-like.[447] Small yards, and a work-room, and a covered shed for working in bad weather, were to be provided for vagrants.[448] For workhouses having a large number of children the Poor Law Board recommended, in addition to the school-rooms, day-rooms, covered play-sheds in their yards, and industrial work-rooms.[449] The staircases were to be of stone; the timber, Baltic fir and English oak; fire escapes were to be provided; these and many other details were laid down, all tending to make the building solid and capacious.[450] There was no mention of ornament, no regard to appearance, no hint that anything might be done to relieve the dead ugliness of the place; but it must be recognised that the Central Authority had, by 1868, travelled far from the low, cheap, homely building which it was recommending thirty years before.[451] Separate dormitories, day-rooms, and yards (apparently not dining- rooms) were required for the aged, able-bodied, children, and sick of each sex, and these were the only divisions laid down as fundamental, but the Circular went on to recommend provision (1) so far as practicable for the sub-division of the able-bodied women into two or three classes with reference to moral character, or behaviour, the previous habits of the inmates, or such other grounds as might seem expedient, and (2) in the larger workhouses for the separate accommodation of the following classes of sick—
- 75. Ordinary sick of both sexes. Lying-in women, with separate labour room. Itch cases of both sexes. Dirty and offensive cases of both sexes. Venereal cases of both sexes. Fever and smallpox cases of both sexes (to be in a separate building with detached rooms). Children (in whose case sex was not mentioned).[452] In the furnishing of the wards the simplicity of 1868 was equally far removed from that of 1835. Ordinary dormitories contained beds 2 feet 6 inches wide, chairs, bells, and gas where practicable. Day- rooms were to have an open fireplace, benches, cupboards (or open shelves, which were preferred), tables, gas, combs, and hairbrushes. A proportion of chairs were to be provided for the aged and infirm; and of the benches, likewise, those for the aged and infirm should have backs, and be of sufficient width for reasonable comfort. In the dining-rooms were to be benches, tables, a minimum of necessary table utensils, and if possible gas and an open fireplace. The sick wards were to be furnished with more care, and with an eye to medical efficiency. It is unnecessary to go into the long and detailed list of the medical appliances which were required. There is even some notice of appearances in a suggestion that cheerful-looking rugs should be placed on the beds, and of comfort in the arm and other chairs for two-thirds of the number of the sick. There were also to be short benches with backs, and (but these only for special cases) even cushions; rocking-chairs for the lying-in wards, and little arm-chairs and rocking-chairs for the children's sick wards.[453] Dr. Smith had further recommended a Bible for each inmate, entertaining illustrated and religious periodicals, tracts and books, games, and a foot valance to the bed to add to the appearance of comfort,[454] These suggestions were not specifically taken up by the Central Authority, but Dr. Smith's report was circulated to the guardians, without comment.[455] We have the beginning, too, between 1863 and 1867, of the improvement of the food, which was regulated in each workhouse by a separate Special
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