Exponential Sums by Irrationality Exponent
Byungchul Cha, Dong Han Kim
TL;DR
This paper studies the exponential sum S(x)= sum_{n <= x} mu(n) e(alpha n) with alpha irrational, linking its size to the irrationality exponent eta of alpha. Using Vaughan's identity to split the sum into Type I and II components and employing Vinogradov-type estimates together with Diophantine approximation via eta, the authors derive eta-dependent bounds. The main results show S(x) = O_eps(x^{4/5+eps}) when 2 <= eta <= 5/2 and S(x) = O_eps(x^{(2 eta-1)/(2 eta)+eps}) when eta >= 5/2, extending Murty and Sankaranarayanan's eta=2 case and avoiding zero-free L-function hypotheses. These bounds contribute to the understanding of Möbius disjointness in rotations and demonstrate that sharper eta-controlled estimates are achievable without assuming zero-free regions for Dirichlet L-functions.
Abstract
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 $η= 2$.