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

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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

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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.

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Results
Below, one can find the calculation results in table form:
Below, one can find the comparative error on a chart:

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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