Exponential Sums by Irrationality Exponent

Abstract

In this article, we give an asymptotic bound for the exponential sum of the Möbius function ${\sum }_{n\le x}\hat{I}¼(n)e(\hat{I}\pm n)$ for a fixed irrational number $\hat{I}\pm \in \mathbb{R}$. This exponential sum was originally studied by Davenport and he obtained an asymptotic bound of $x(\log x{)}^{-A}$ for any $A\ge 0$. Our bound depends on the irrationality exponent $\hat{I}\cdot$ of $\hat{I}\pm$. If $\hat{I}\cdot \le 5/2$, we obtain a bound of $x^{4/5+\epsilon }$ and, when $\hat{I}\cdot \ge 5/2$, our bound is $x^{(2\hat{I}\cdot -1)/2\hat{I}\cdot +\epsilon }$. This result extends a result of Murty and Sankaranarayanan, who obtained the same bound in the case $\hat{I}\cdot =2$.

5 pages