The distribution of the maximum of character sums

TL;DR

This paper provides explicit bounds on the moments of character sums, leading to a detailed understanding of the distribution of their maximum magnitude, with results showing rapid decay in the tail probabilities.

Contribution

It refines previous estimates of Montgomery and Vaughan by deriving explicit bounds on moments and character sum distributions.

Findings

Explicit bounds on moments of character sums

Distribution of maximal character sum magnitude with double exponential tail decay

Enhanced understanding of character sum behavior

Abstract

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.