Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions
K. Venkatasubbareddy
TL;DR
This work studies the higher-moment averages of the Fourier coefficients of symmetric power $L$-functions attached to a primitive holomorphic cusp form. It develops a framework based on the Dirichlet series $L_{l,j}(s)$, obtained from the $l$th power of $\lambda_{\mathrm{sym}^j f}(n)$, and decomposes it into a product of zeta powers and symmetric-power $L$-functions of degree $D=(j+1)^l$. The authors prove a general asymptotic: for $lj$ even, $\\sum_{n\\le x} \\lambda_{ {\\rm{sym}}^j}^l(n)= x P_{d_{lj/2}-1}(\\log x) + O(x^{\\theta_{l,j}+\\varepsilon})$, and for $lj$ odd, a pure power-saving bound $O(x^{\\theta_{l,j}+\\varepsilon})$, with explicitly given $\\theta_{l,j}$ depending on $D$, $d_m$, and $e_m$, thus improving previous exponents. A special treatment is provided for $l=j=2$. The results advance understanding of higher-moment behavior of symmetric power Fourier coefficients and have potential implications for refined bounds in the analytic theory of $L$-functions. Methods include Perron’s formula, contour integration, and mean-value bounds for families of $L$-functions, combined to yield sharp main terms and quantified error terms.
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 $λ_{\rm{sym}^jf}(n)$ 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 $λ_{\rm{sym}^jf}(n)$ and their higher powers, and many researchers in the literature have studied the sum \begin{equation*} \sum_{n\leq x} λ_{\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 $lj\geq 4$.