Around Wilson's theorem
Number Theory
2018-09-11 v1
Abstract
We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in the Lebesgue measure s(n,x) is equivalent to n/2 when n tends to infinity, while for almost all x in the Baire sense, 1/2 is a limit point of the ratio of s(n,x) to the number of primes less or equal to n.
Cite
@article{arxiv.1809.02832,
title = {Around Wilson's theorem},
author = {Alain Connes},
journal= {arXiv preprint arXiv:1809.02832},
year = {2018}
}
Comments
6 pages 3 Figures, Journal of Number Theory 2018