Long large character sums

TL;DR

This paper establishes a new lower bound for large character sums over specific ranges, improving previous results and aligning with expected maximal values as the range parameter diminishes.

Contribution

It provides a novel lower bound for large character sums in a wide range, extending prior work by Granville and Soundararajan.

Findings

New lower bound for character sums when x = q / (log q)^B

Improved understanding of maximal character sums for large ranges

Results recover expected maximal values as B approaches zero

Abstract

In this paper, we prove a lower bound for $\underset{\chi \neq \chi_0}{\max}\bigg|\sum_{n\leq x} \chi(n)\bigg|$, when $x= \frac{q}{(\log q)^B}$. This improves on a result of Granville and Soundararajan for large character sums when the range of summation is wide. When $B$ goes to zero, our lower bound recovers the expected maximal value of character sums for most characters.