Average sizes of mixed character sums

TL;DR

This paper establishes that the average size of mixed character sums is approximately proportional to the square root of x for irrational theta under certain conditions, contrasting with smaller averages for rational theta.

Contribution

It proves a new bound on the average size of mixed character sums for irrational theta, extending understanding beyond rational cases using Diophantine conditions.

Findings

Average size of sums is on the order of √x for irrational θ.

For rational θ, the average size is o(√x).

Quadratic Diophantine equations are crucial in the analysis.

Abstract

We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} \chi(n) e(n\theta) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine condition, where $\chi$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\le r$. In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $\theta$. Certain quadratic Diophantine equations play a key role in the present paper.