# Moments of character sums to composite modulus

## TL;DR

This paper advances the understanding of character sums over composite moduli by extending techniques from prime moduli, leading to improved bounds and progress towards removing the cubefree restriction in the Burgess bound.

## Contribution

It introduces a method to estimate high order moments of character sums for composite moduli, extending existing techniques from prime moduli, and improves existing bounds such as Norton's estimate.

## Key findings

- Progress towards removing the cubefree restriction in the Burgess bound.
- Extension of techniques from prime to composite moduli for character sum estimates.
- Improved bounds on character sums to composite moduli.

## Abstract

In this paper we consider the problem of estimating character sums to composite modulus and obtain some progress towards removing the cubefree restriction in the Burgess bound. Our approach is to estimate high order moments of character sums in terms of solutions to congruences with Kloosterman fractions and we deal with this problem by extending some techniques of Bourgain, Garaev, Konyagin and Shparlinski and Bourgain and Garaev from the setting of prime modulus to composite modulus. As an application of our result we improve an estimate of Norton.