Some results on divisor problems related to cusp forms

Abstract

Let $\lambda _{f}(n)$ be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with $\lambda _{f}(n)$ over some special sequences. More precisely, for any fixed integer $k\ge 2$ and $j\in \{1,2,3,4\},$ we are interested in the following sums

$$\begin{aligned} S_{k}(x,j):=\sum _{n\le x}\lambda _{k,f}(n^{j})=\sum _{n\le x}\sum _{n=n_{1}n_{2}\cdots n_{k}}\lambda _{f}(n_{1}^{j})\lambda _{f}(n_{2}^{j})\cdots \lambda _{f}(n_{k}^{j}). \end{aligned}$$