The frequency and the structure of large character sums

Abstract

Let $M(χ)$ denote the maximum of $|\sum_{n\le N}χ(n)|$ for a given non-principal Dirichlet character $χ\pmod q$, and let $N_χ$ denote a point at which the maximum is attained. In this article we study the distribution of $M(χ)/\sqrt{q}$ as one varies over characters $\pmod q$, where $q$ is prime, and investigate the location of $N_χ$. We show that the distribution of $M(χ)/\sqrt{q}$ 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, $M(χ)\ge |\sum_{n\le q/2}χ(n)| = \frac{|2-χ(2)|}π\sqrt{q} |L(1,χ)|$, 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 $\frac{\sqrt{q}}π |L(1,χ)|$.