Short character sums for composite moduli

TL;DR

This paper presents improved estimates for short character sums over composite moduli with small prime factors, enhancing bounds relevant to number theory and L-functions.

Contribution

It introduces new bounds on short character sums for composite moduli, surpassing previous results by Graham-Ringrose and Gallagher-Iwaniec under certain conditions.

Findings

Improved bounds on short character sums for composite moduli.

Enhanced zero-free regions for Dirichlet L-functions.

Refinements to Pólya and Vinogradov inequalities.

Abstract

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square free moduli and also on the result due to Gallagher and Iwaniec when the core $q'=\prod_{p|q}p$ of the modulus $q$ satisfies $\log q'\sim \log q$. Some applications to zero free regions of Dirichlet L-functions and the $\rm{P\acute{o}lya}$ and Vinogradov inequalities are indicated.