Better approximation for π(x)
First date: 17/09/2013 Author: Chris De Corte E-mail:
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Abstract In this document, we will show that: ߨሺݔሻ =
ݔ 4 . [1 − ඨ1 − ]−7 2 lnሺݔሻ
might be a better approximation for the prime-counting function than ߨሺݔሻ = ݔ/[lnሺݔሻ − 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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Conclusion: 1. Our formula gives better results
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
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