The frequency and the structure of large character sums

Let M(?) denote the maximum of |sigma(n <= N)?(n)| given non-principal Dirichlet character ? modulo q, and let N-? denote a point at which the maximum is attained. In this article we study the distribution of M(?)/root q as one varies over characters modulo q, where q is prime, and investigate the location of N-?. We show that the distribution of M(?)/root q converges weakly to a universal distribution Phi, uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for Phi's tail. Almost all ? for which M(?) is large are odd characters that are 1-pretentious. Now, M(?) >= |Sigma(n <= q/2)?(n)| = (|2 - ?(2)|/pi|)root q|L(1, ?)|, and one knows how often the latter expression is large, which has been how earlier lower bounds on Phi were mostly proved. We show, though, that for most ? with M(?) large, N-? is bounded away from q/2, and the value of M(?) is little larger than (root q/pi)|L(1,?)|.