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arXiv · 2506.12845

Bounded Exponential Sums with Multiplicative Coefficients

Pierre-Alexandre Bazin·Ihor Pylaiev·Fred Tyrrell
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Abstract

We investigate when the exponential sum $S_f(x,\alpha) := \sum_{n\le x}f(n)\mathrm{e}(n\alpha)$ is bounded, for a multiplicative function $f$ and $\alpha\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,\alpha)$ is bounded only when $f$ is very close to a twisted Dirichlet character $\chi(n)n^{it}$. 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 $\alpha$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $\alpha$.

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Pierre-Alexandre Bazin, Ihor Pylaiev, Fred Tyrrell. 2025-06-15. Bounded Exponential Sums with Multiplicative Coefficients. https://doi.org/10.1112/mtk.70117

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