The frequency and the structure of large character sums

TL;DR

This paper analyzes the distribution and structure of large character sums for Dirichlet characters, revealing their typical behavior, the location of maximum sums, and establishing a universal distribution with exponential tail decay.

Contribution

It introduces a universal distribution for normalized large character sums, studies the location of their maxima, and connects these sums to $L$-values, providing new insights into their typical size and structure.

Findings

Distribution of $M( ext{chi})/\sqrt{q}$ converges to a universal distribution $$.

Most large sums occur for odd, 1-pretentious characters.

The maximum sum $M( ext{chi})$ is closely related to $|L(1, ext{chi})|$ and is typically bounded away from $q/2$.

Abstract

Let $M(\chi)$ denote the maximum of $|\sum_{n\le N}\chi(n)|$ for a given non-principal Dirichlet character $\chi \pmod q$, 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 $\pmod q$, 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 $\frac{\sqrt{q}}{\pi} |L(1,\chi)|$.