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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.06726 |
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Table of 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 $η= 2$.