On the sum of the twisted Fourier coefficients of Maass forms by M\"obius function

Abstract

In this paper, we study non-trivial upper bounds for the sum $\sum \limits_{n \in S} |\lambda_f(n)|$ where $f$ is a normalized Maass eigencusp form for the full modular group, $\lambda_f(n)$ is the $n$th normalized Fourier coefficient of $f$ and $S$ is a proper subset of positive integers in $[1,x]$ with certain properties.