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Better prime counting formula

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In this document, we will show a better approximation for the prime-counting function proposed by Bernhard Riemann.

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Better prime counting formula

1. 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. 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. 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. 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