The maximum size of short character sums

TL;DR

This paper establishes new lower bounds on the maximum size of character sums in specific ranges, using advanced resonance methods and properties of friable integers to improve previous results.

Contribution

It introduces improved lower bounds for character sums in certain ranges by applying resonance techniques and analyzing the distribution of friable integers.

Findings

Enhanced lower bounds for character sums in specific ranges.

Application of resonance method to large value problems.

Refined analysis of friable integers in short intervals.

Abstract

In the present note, we prove new lower bounds on large values of character sums $\Delta(x,q):=\max_{\chi \neq \chi_0} \vert \sum_{n\leq x} \chi(n)\vert$ in certain ranges of $x$. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of $L$- functions, we improve some results of Hough in the range $\log x = o(\sqrt{\log q})$. Our results are expressed using the counting function of $y$- friable integers less than $x$ where we improve the level of smoothness $y$ for short intervals.