Bounded Exponential Sums with Multiplicative Coefficients

Abstract

We investigate when the exponential sum $S_f(x,\hat{I}\pm ):={\sum }_{n\le x}f(n)\mathrm{e}(n\hat{I}\pm )$ is bounded, for a multiplicative function $f$ and $\hat{I}\pm \in \mathbb{R}$. We show that under natural assumptions, $S_f(x,\hat{I}\pm )$ is bounded only when $f$ is very close to a twisted Dirichlet character $\ddot{I}‡(n)n^{it}$. We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when $f$ is completely multiplicative and $\hat{I}\pm$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $\hat{I}\pm$.

29 pages. Revised version with an additional author and main theorems strengthened. Sections 4 and 5 replaced, small changes in other sections