Type: Seminar

Distribution: World

Expiry: 18 Oct 2019

CalTitle1: Prime number races

Auth: leo@121.213.25.198 (ltzo2369) in SMS-WASM

This talk is a survey of ``prime number races". Around 1850, Chebyshev noticed 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+1,qn+bn,qn+c,â€¦ for our favorite constants q,a,b,c,â€¦ and try to figure out which forms are ``preferred" over the others---not to mention figuring out what, precisely, being ``preferred" means. All of these ``races’’ are related to the function Ï€(x) that counts the number of primes up to x, which has both an asymptotic formula with a wonderful proof and an associated ``race’’ of its own; and the attempts to analyze these races are closely related to the Riemann hypothesis---the most famous open problem in mathematics. We describe these phenomena, in an accessible way, in greater detail; we provide examples of computations that demonstrate the ``preferences’’ described above; and we explain the efforts that have been made at understanding the underlying mathematics.