Investigation of the Characteristics of the Zeros of the Riemann Zeta Function in the Critical Strip Using Implicit Function Properties of the Real and Imaginary Components of the Dirichlet Eta function
- 1. Why the Dirichlet eta function and so the Riemann zeta Function have only zeros with real part = ½. A Proof of the Riemann Hypothesis A. A. Logan February 2019
- 2. Overview • Abstract • Zeta function observations (when Re(zeta) = Im(zeta)). • Dirichlet eta function - Implicit function of Real component = Imag component – derivative of value of real component always nonzero so only one zero for one value of imag part of s (using harmonic addition theorem). • Riemann xi function – implications. • Nb xi(s) = xi(1-s). • Conclusions, link to paper and contact details.
- 3. Abstract • This paper investigates the characteristics of the zeros of the Riemann zeta function (of s) in the critical strip by using the Dirichlet eta function, which has the same zeros. • The characteristics of the implicit functions for the real and imaginary components when those components are equal are investigated and it is shown that the function describing the value of the real component when the real and imaginary components are equal has a derivative that does not change sign along any of its individual curves - meaning that each value of the imaginary part of s produces at most one zero. • Combined with the fact that the zeros of the Riemann xi function are also the zeros of the zeta function and xi(s) = xi(1-s), this leads to the conclusion that the Riemann Hypothesis is true.
- 5. Zeta function components - detail
- 6. Zeta function observations • • Different symmetries of Re(zeta) and Im(zeta). • Path of Re(zeta)=Im(zeta) example (slope). • Value of Re(zeta) when Re(zeta)=Im(zeta). • Apparent single valued curve.
- 7. Riemann Zeta Function and Dirichlet Eta Function Note Zeta converges absolutely for s>1 Eta converges uniformly (not absolutely) for s>0 Eta zeroes in critical strip same as Zeta zeroes.
- 8. Dirichlet eta function Extracting real and imaginary parts for one term: Real part: Imag part:
- 10. Differentiating Implicit Function(1) Implicit function: Differentiating: Substituting into exp for real component:
- 11. Differentiating Implicit Function(2) Similarly: And: Nb this describes the same function as the last slide.
- 12. Harmonic addition Theorem(1) Using partial sums and substituting in (exp 1): And:(exp 2) Also (exp3): And: (exp4)
- 13. Harmonic addition Theorem(2) Beta does not equal zero Csc has no zeros Exp2 is the derivative of the real part of eta (when re(eta)=imag(eta)) and is always positive (or always negative) for all a, sigma (except where undefined). The same holds for Exp4 (undefined at different points). This means only one zero for each curve segment (or max one zero for each value of a)
- 14. Riemann xi Function • xi function - Power Series • xi function zeros • See above slide for only one root per value of a - so sigma = ½.
- 15. Conclusions • Eta(s) has same roots as zeta(s) • Eta(s) real component value when Re(eta) = Im(eta) – shows always non-zero derivative. • eta(s) only 1 zero per fixed imag part of s. • xi(s) = xi(1-s) means that root must be at real part of s = ½: Riemann Hypothesis confirmed. • http://vixra.org/abs/1802.0124 • andrewalogan@gmail.com