The distribution of large quadratic character sums and applications

TL;DR

This paper studies the maximum distribution of quadratic character sums, improves existing bounds, and explores implications for prime distributions and Legendre symbol sums using quadratic large sieve techniques.

Contribution

It introduces a new approach based on quadratic large sieve to analyze character sums, extending results to broader families and improving bounds.

Findings

Improved bounds on maximum quadratic character sums.

Evidence supporting the optimality of Bateman and Chowla's Omega result.

Almost all large Legendre symbol sums occur for primes congruent to 3 mod 4.

Abstract

In this paper, we investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $|d|\leq x$. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to $x$, and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to $3$ modulo $4$. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of non-principal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols.