Joint Colloquium: Martin -- Prime number races

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,… 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.  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.