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 $\chi (\mathrm{mod}q)$, 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 $(\mathrm{mod}q)$, 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 $\frac{\sqrt{q}}{\pi }|L(1,\chi )|$.

58 pages, 4 figures, 1 table. Minor changes, and small improvement of Theorem 2.2. Final version, to appear in J. Eur. Math. Soc

References in corpus (1)

• Averages of character sums

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