The frequency and the structure of large character sums

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