The frequency and the structure of large character sums

Abstract

Let $M(\chi )$ denote the maximum of $|{\sum }_{n\le N}\chi (n)|$ for a given non-principal Dirichlet character $\chi$ modulo $q$, and let $N_{\chi }$ denote a point at which the maximum is attained. In this article we study the distribution of $M(\chi )/\sqrt{q}$ as one varies over characters modulo $q$, where $q$ is prime, and investigate the location of $N_{\chi }$. We show that the distribution of $M(\chi )/\sqrt{q}$ 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 $\chi$ for which $M(\chi )$ is large are odd characters that are 1-pretentious. Now, $M(\chi )\ge |{\sum }_{n\le q/2}\chi (n)|=\frac{|2-\chi (2)|}{\pi }\sqrt{q}|L(1,\chi )|$, 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 $\chi$ with $M(\chi )$ large, $N_{\chi }$ is bounded away from $q/2$, and the value of $M(\chi )$ is little bit larger than $\frac{\sqrt{q}}{\pi }|L(1,\chi )|$.