Prime Number Races

Abstract

This talk is a survey of “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 than there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, 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 – not to mention figuring out what, precisely, being “preferred” means. We describe these phenomena in greater detail and explain the efforts that have been made at understanding them. This talk should be accessible to graduate students. 1272547773