Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions

TL;DR

This paper investigates the average behavior of Fourier coefficients of symmetric power L-functions associated with modular forms, extending known results to broader cases for higher moments and symmetric powers.

Contribution

It improves and generalizes existing results on sums of powers of Fourier coefficients of symmetric power L-functions for various indices.

Findings

Extended the range of $l$ and $j$ for which sum estimates are known.

Provided new bounds and asymptotic formulas for these sums.

Enhanced understanding of the distribution of Fourier coefficients in symmetric power L-functions.

Abstract

For an even integer $k\geq 2$, let $f$ be a primitive holomorphic cusp form of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ and let $\lambda_{{\rm{sym}}^jf}(n)$ denote the $n^\text{th}$ normalized Fourier coefficient of the $j^{\text{th}}$ symmetric power $L$-function $L(s,{\rm{sym}}^j f)$. It has been an interesting problem to study the average behaviour of $\lambda_{{\rm{sym}}^jf}(n)$ and their higher powers, and many researchers in the literature have studied the sum \begin{equation*} \sum_{n\leq x} \lambda_{{\rm{sym}}^j}^l(n), \end{equation*} for various values of $l$ and $j$. In this paper, we improve and generalize previously known results concerning the sum above for positive integers $l$ and $j$ such that $lj\geq 4$.