Mixed incomplete character sums of rational functions with smooth moduli

TL;DR

This paper improves bounds on character sums over smooth moduli, showing that the smoothness condition can be relaxed to N^{1-ε}, with various generalizations and optimal results.

Contribution

It extends Graham and Ringrose's bounds to less smooth moduli, achieving the best possible results in several aspects.

Findings

Bound on character sums with less smooth moduli

Relaxation of smoothness condition to N^{1-ε}

Applications to number theory problems

Abstract

Let $\chi=\chi_q$ be a primitive character mod $q$ and fix $\Delta>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} \chi(n)$ in intervals of length $N=q^\Delta$ whenever $q$ is squarefree and is sufficiently smooth. Here we show that the smoothness parameter can be taken to be $N^{1-\epsilon}$. We also discuss various generalizations and applications, obtaining best possible results in several aspects.