Better prime counting formula
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This document proposes a new approximation for the prime-counting function, suggesting that the formula π(x) = x^2 [1 - (1 - 4 ln(x))] - 7 is more accurate than the Riemann-influenced formula π(x) = x/ln(x) - 1. The research involved calculations performed using Microsoft Excel, which demonstrated that the new formula yields better results. The author expresses gratitude to various individuals and acknowledges the efforts of the publisher and staff.
![Author : chrisdecorte@yahoo.com Page 2
Abstract
In this document, we will show that:
ߨሺݔሻ =
ݔ
2
. [1 − ඨ1 −
4
lnሺݔሻ
] − 7
might be a better approximation for the prime-counting function than ߨሺݔሻ =
[/ݔlnሺݔሻ − 1] proposed by Bernhard Riemann [1].
Key-words
prime number theorem (PNT), prime-counting function, asymptotic law of
distribution, Riemann hypothesis, Clay Mathematics.
Introduction
The following document originated during our study of primes and the reading
about the Riemann hypothesis [2,3].
We were baffled by the fact that the young Riemann had found such a complex
formula as a proposition for to the prime-counting function.
We were curious to find a better formula.
Methods & Techniques
We used Microsoft Excel to do our calculations.](https://image.slidesharecdn.com/betterprimecountingformula-130727082052-phpapp01/85/Better-prime-counting-formula-2-320.jpg)
Better prime counting formula
- 1. Author : chrisdecorte@yahoo.com Page 1 Better approximation for π(x) II Author: Chris De Corte KAIZY BVBA Beekveldstraat 22 bus 1 9300 Aalst Belgium Tel: +32 495/75.16.40 E-mail: chrisdecorte@yahoo.com
- 2. Author : chrisdecorte@yahoo.com Page 2 Abstract In this document, we will show that: ߨሺݔሻ = ݔ 2 . [1 − ඨ1 − 4 lnሺݔሻ ] − 7 might be a better approximation for the prime-counting function than ߨሺݔሻ = [/ݔlnሺݔሻ − 1] proposed by Bernhard Riemann [1]. Key-words prime number theorem (PNT), prime-counting function, asymptotic law of distribution, Riemann hypothesis, Clay Mathematics. Introduction The following document originated during our study of primes and the reading about the Riemann hypothesis [2,3]. We were baffled by the fact that the young Riemann had found such a complex formula as a proposition for to the prime-counting function. We were curious to find a better formula. Methods & Techniques We used Microsoft Excel to do our calculations.
- 3. Author : chrisdecorte@yahoo.com Page 3 Results Below, one can find the calculation results in table form: Below, one can find the comparative error on a chart:
- 4. Author : chrisdecorte@yahoo.com Page 4 Discussions: Conclusion: 1. Our formula gives better results Acknowledgements I would like to thank this publisher, his professional staff and his volunteers for all the effort they take in reading all the papers coming to them and especially I would like to thank this reader for reading my paper till the end. I would like to thank Jens Kruse Andersen, David Eppstein and Renaud Lifchitz for taking the time to react to my mails. References 1. https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_ Magnitude 2. http://en.wikipedia.org/wiki/Prime_number_theorem 3. https://en.wikipedia.org/wiki/Riemann_hypothesis