On an exponential sum related to the Möbius function

Abstract

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

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