A note on average behaviour of the Fourier coefficients of $j$th symmetric power $L$-function over certain sparse sequence of positive integers

Wang, Youjun. "A note on average behaviour of the Fourier coefficients of $j$th symmetric power $L$-function over certain sparse sequence of positive integers." Czechoslovak Mathematical Journal 74.2 (2024): 623-636. <http://eudml.org/doc/299376>.

@article{Wang2024,
abstract = {Let $j\ge 2$ be a given integer. Let $H_\{k\}^\{*\}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\ge 2$ for the full modulo group $\{\rm SL\}(2,\mathbb \{Z\})$. For $f\in H_\{k\}^\{*\}$, denote by $\lambda _\{\{\rm sym\}^\{j\}f\}(n)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s, \{\rm sym\}^\{j\}f)$) attached to $f$. We are interested in the average behaviour of the sum \[ \sum \_\{n=a\_\{1\}^\{2\}+a\_\{2\}^\{2\}+a\_\{3\}^\{2\}+a\_\{4\}^\{2\}+a\_\{5\}^\{2\}+a\_\{6\}^\{2\}\le x \atop (a\_\{1\},a\_\{2\},a\_\{3\},a\_\{4\},a\_\{5\},a\_\{6\})\in \mathbb \{Z\}^\{ 6\}\} \lambda \_\{\{\rm sym\}^\{j\}f\}^\{2\}(n), \] where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).},
author = {Wang, Youjun},
journal = {Czechoslovak Mathematical Journal},
keywords = {cusp form; Fourier coefficient; symmetric power $L$-function},
language = {eng},
number = {2},
pages = {623-636},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on average behaviour of the Fourier coefficients of $j$th symmetric power $L$-function over certain sparse sequence of positive integers},
url = {http://eudml.org/doc/299376},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Wang, Youjun
TI - A note on average behaviour of the Fourier coefficients of $j$th symmetric power $L$-function over certain sparse sequence of positive integers
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 623
EP - 636
AB - Let $j\ge 2$ be a given integer. Let $H_{k}^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\ge 2$ for the full modulo group ${\rm SL}(2,\mathbb {Z})$. For $f\in H_{k}^{*}$, denote by $\lambda _{{\rm sym}^{j}f}(n)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s, {\rm sym}^{j}f)$) attached to $f$. We are interested in the average behaviour of the sum \[ \sum _{n=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\le x \atop (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in \mathbb {Z}^{ 6}} \lambda _{{\rm sym}^{j}f}^{2}(n), \] where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).
LA - eng
KW - cusp form; Fourier coefficient; symmetric power $L$-function
UR - http://eudml.org/doc/299376
ER -