Exponential Sums by Irrationality Exponent
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910913852866560 |
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| author | Cha, Byungchul Kim, Dong Han |
| author_facet | Cha, Byungchul Kim, Dong Han |
| contents | In this article, we give an asymptotic bound for the exponential sum of the Möbius function $\sum_{n \le x} μ(n) e(αn)$ for a fixed irrational number $α\in\mathbb{R}$. This exponential sum was originally studied by Davenport and he obtained an asymptotic bound of $x(\log x)^{-A}$ for any $A\ge0$. Our bound depends on the irrationality exponent $η$ of $α$. If $η\le 5/2$, we obtain a bound of $x^{4/5 + \varepsilon}$ and, when $η\ge 5/2$, our bound is $x^{(2η-1)/2η+ \varepsilon}$. This result extends a result of Murty and Sankaranarayanan, who obtained the same bound in the case $η= 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential Sums by Irrationality Exponent Cha, Byungchul Kim, Dong Han Number Theory 11L07 (Primary), 37E10 (Secondary) In this article, we give an asymptotic bound for the exponential sum of the Möbius function $\sum_{n \le x} μ(n) e(αn)$ for a fixed irrational number $α\in\mathbb{R}$. This exponential sum was originally studied by Davenport and he obtained an asymptotic bound of $x(\log x)^{-A}$ for any $A\ge0$. Our bound depends on the irrationality exponent $η$ of $α$. If $η\le 5/2$, we obtain a bound of $x^{4/5 + \varepsilon}$ and, when $η\ge 5/2$, our bound is $x^{(2η-1)/2η+ \varepsilon}$. This result extends a result of Murty and Sankaranarayanan, who obtained the same bound in the case $η= 2$. |
| title | Exponential Sums by Irrationality Exponent |
| topic | Number Theory 11L07 (Primary), 37E10 (Secondary) |
| url | https://arxiv.org/abs/2504.06726 |