Average sizes of mixed character sums

Abstract

We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} χ(n) e(nθ) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $θ$ satisfying a weak Diophantine condition, where $χ$ 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 $θ$. Certain quadratic Diophantine equations play a key role in the present paper.