On the higher power moments of Dirichlet coefficients of symmetric power L-functions and its applications

Abstract

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Let $\lambda_{\textrm{sym}^{j}f}(n)$ denotes the $n$th normalized coefficients of the Dirichlet expansion  of the $j$th symmetric power $L$-function.   In this paper, we are interested in the average behaviour of higher moments of $\lambda_{\textrm{sym}^{j}f}(n)$ for $j\geq 2$, which refines the previous results in this direction. As an application, we also consider the number of sign changes of the sequence $\{\lambda_{\textrm{sym}^{j}f}(n)\}$  for $j\geq 3$ in the interval $(x,2x]$.