Around Wilson's theorem

TL;DR

This paper explores a series involving sine functions and Gamma functions, revealing connections to prime counting via Wilson's theorem and analyzing the series' behavior for different values of x.

Contribution

It introduces a novel series s(n,x) linked to prime counting and demonstrates its properties using Wilson's theorem and measure theory.

Findings

s(n, Pi/2) equals the number of primes ≤ n

For almost all x, s(n,x) ~ n/2 as n→∞

1/2 is a limit point of s(n,x)/π(n) in the Baire sense

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.