2018/09/08 by Alain Connes, Connes, Alain
Mathematics · #11A41 #Analytic Number Theory Research #Combinatorics #Discrete mathematics #FOS: Mathematics #History and Theory of Mathematics #Infinity #Integer (computer science) #Lebesgue integration #Lebesgue measure #Limit (mathematics) #Mathematical analysis #Mathematics #Mathematics and Applications #Number Theory (math.NT) #Pi #Product (mathematics) #Square (algebra) #math.NT #msc:11A41
paper · pdf · doi:10.48550/arxiv.1809.02832
6 pages 3 Figures, Journal of Number Theory 2018
arxiv created 2018/09/08 · openalex publication_date 2018/09/08 · arxiv updated 2018/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.