Average sizes of mixed character sums

Abstract

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

14 pages; final version