On the average behavior of the Fourier coefficients of $j$th symmetric power $L$-function over certain sequences of positive integers

DOI10.21136/CMJ.2023.0348-22arXiv2206.01491OpenAlexW4367181239MaRDI QIDQ6175176

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Publication date: 17 August 2023

Published in: Czechoslovak Mathematical Journal (Search for Journal in Brave)

Abstract: In this paper, we investigate the average behavior of the $n^{th}$ normalized Fourier coefficients of the $j^{th}$ ($jgeq2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,sy{m}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,mathbbZ)$ over a certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum ${\displaystyle }$sum_{stackrel{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}leq {x}}{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})inmathbb{Z}^{6}}}lambda^{2}_{sym^{j}f}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),${\displaystyle }$ where $x$ is sufficiently large, and ${\displaystyle }$L(s,sym^{j}f):=sum_{n=1}^{infty}dfrac{lambda_{sym^{j}f}(n)}{n^{s}}.${\displaystyle }$ When $j=2$, the error term which we obtain, improves the earlier known result.

Full work available at URL: https://arxiv.org/abs/2206.01491

zbMATH Keywords

Hölder's inequalityholomorphic cusp formnonprincipal Dirichlet character\(j\)th symmetric power \(L\)-function

Mathematics Subject Classification ID

(zeta (s)) and (L(s, chi)) (11M06) Fourier coefficients of automorphic forms (11F30) Holomorphic modular forms of integral weight (11F11)

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