2004/08/24 by Andrew Granville, Greg Martin, Granville, Andrew +1 · 1 citation
Mathematics · #11N13 #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT #msc:11N13
paper · pdf · doi:10.48550/arxiv.math/0408319
a survey article; 38 pages (including appendices by Giuliana Davidoff and Michael Guy), 10 tables, 6 figures
arxiv created 2004/08/24 · openalex publication_date 2004/08/24 · arxiv updated 2009/12/01 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, with primes of the form 10n+3/10n+7 and 10n+1/10n+9, and many others besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c, >... for our favorite constants q, a, b, c, ... and try to figure out which forms are "preferred" over the others. In this paper, we describe these phenomena in greater detail and explain the efforts that have been made at understanding them.