On an exponential sum related to the M\"{o}bius function

TL;DR

This paper investigates upper bounds for exponential sums involving the Möbius function, improving classical results and contributing to understanding their behavior in analytic number theory.

Contribution

It provides improved upper bounds for the sum involving the Möbius function and exponential terms, refining previous results by Baker and Harman.

Findings

Enhanced bounds for the exponential sum S(x,α)

Refinement of classical results in the literature

Implications for the distribution of the Möbius function

Abstract

Let $\mu(n)$ be the M\"{o}bius function and $e(\alpha)=e^{2\pi i\alpha}$. In this paper, we study upper bounds of the classical sum $$S(x,\alpha):=\sum_{1\leq n\leq x}\mu(n)e(\alpha n).$$ We can improve some classical results of Baker and Harman \cite{BH}.