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 modulo q , and let N_χ 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 modulo q , where q is prime, and investigate the location of N_χ . 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)|}π √q |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, N_χ is bounded away from q/2 , and the value of M(χ) is little bit larger than {√q}{π} |L(1,χ)| .