2011 CMS Winter Meeting Delta Chelsea Hotel, Toronto, December 10 - 12, 2011 www.cms.math.ca//Events/winter11

Doctoral Prize

YOUNESS LAMZOURI, University of Illinois at Urbana-Champaign
Prime number races  [PDF]

Although the primes are equidistributed in arithmetic progressions, it has been noted that certain residue classes tend to contain more primes in initial intervals of the positive integers. This phenomenon was first observed by Chebyshev in 1853. Since that time, races'' between primes in arithmetic progressions have been extensively studied. A prime number race $\{q;a_1,\dots,a_r\}$ is a game with $r$ players, where that at time $t$, the score of the $j$-th player is the number of primes less than $t$ that are congruent to $a_j$ modulo $q$. In this lecture, I will review the history of this subject, and discuss recent progress towards understanding the origin of Chebyshev's bias and its generalizations. To study these questions we will require deep information on the zeros of certain analytic functions called Dirichlet $L$-functions.