Bounded Exponential Sums with Multiplicative Coefficients
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908844943212544 |
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| author | Bazin, Pierre-Alexandre Pylaiev, Ihor Tyrrell, Fred |
| author_facet | Bazin, Pierre-Alexandre Pylaiev, Ihor Tyrrell, Fred |
| contents | We investigate when the exponential sum $S_f(x,α) := \sum_{n\le x}f(n)\mathrm{e}(nα)$ is bounded, for a multiplicative function $f$ and $α\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,α)$ is bounded only when $f$ is very close to a twisted Dirichlet character $χ(n)n^{it}$. 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 $α$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded Exponential Sums with Multiplicative Coefficients Bazin, Pierre-Alexandre Pylaiev, Ihor Tyrrell, Fred Number Theory 11L07 (Primary) 11N64, 11J71 (Secondary) We investigate when the exponential sum $S_f(x,α) := \sum_{n\le x}f(n)\mathrm{e}(nα)$ is bounded, for a multiplicative function $f$ and $α\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,α)$ is bounded only when $f$ is very close to a twisted Dirichlet character $χ(n)n^{it}$. 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 $α$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $α$. |
| title | Bounded Exponential Sums with Multiplicative Coefficients |
| topic | Number Theory 11L07 (Primary) 11N64, 11J71 (Secondary) |
| url | https://arxiv.org/abs/2506.12845 |