Higher symmetric power $L$-functions and their Fourier coefficients

TL;DR

This paper derives an asymptotic formula with an improved error term for the sum of squared Fourier coefficients of symmetric power L-functions over integers representable as sums of six squares.

Contribution

It provides a new asymptotic formula with a refined error term for sums involving Fourier coefficients of symmetric power L-functions over specific integer representations.

Findings

Asymptotic formula for the sum of squared Fourier coefficients established.

Improved error term achieved in the asymptotic estimate.

Results apply to sums over integers represented as sums of six squares.

Abstract

Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j }f}(n)$ for the $n^\textit{th}$ normalized Fourier coefficient of $L(s,{\rm{sym}}^j f)$. In this article, we establish an asymptotic formula for the sum $$\begin{equation} \sum_{\substack{n=a_1^2+a_2^2+\ldots+a_6^2\leq x\\ \left(a_1,a_2,\ldots, a_6\right)\in \mathbb{Z}^6}} \lambda_{{\rm{sym}}^j f}^2(n), \end{equation}$$ with an improved error term.