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*Prime number races*

Greg Martin
University of British Columbia

Thursday, May 6, 2010
007 Kemeny Hall, 4 pm
Tea 3:30 pm, 300 Kemeny Hall

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 $qn+a, qn+b, qn+c, \dots$ for our favorite constants $q, a, b, c,\dots$ 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 will be accessible to graduate students.