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

Sharma, Anubhav, and Sankaranarayanan, Ayyadurai. "On the average behavior of the Fourier coefficients of $j$th symmetric power $L$-function over certain sequences of positive integers." Czechoslovak Mathematical Journal 73.3 (2023): 885-901. <http://eudml.org/doc/299113>.

@article{Sharma2023,
abstract = {We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j \ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,\{\rm sym\}^\{j\}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb \{Z\})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum \[ S\_j^*:= \sum \_\{a\_\{1\}^\{2\}+a\_\{2\}^\{2\}+a\_\{3\}^\{2\}+a\_\{4\}^\{2\}+a\_\{5\}^\{2\}+a\_\{6\}^\{2\}\le x (a\_\{1\},a\_\{2\},a\_\{3\},a\_\{4\},a\_\{5\},a\_\{6\})\in \mathbb \{Z\}^\{6\}\} \lambda ^\{2\}\_\{\{\rm sym\}^\{j\}f\}(a\_\{1\}^\{2\}+a\_\{2\}^\{2\}+a\_\{3\}^\{2\}+a\_\{4\}^\{2\}+a\_\{5\}^\{2\}+a\_\{6\}^\{2\}), \] where $x$ is sufficiently large, and \[ L(s,\mathrm \{sym\}^\{j\}f):=\sum \_\{n=1\}^\{\infty \}\frac\{\lambda \_\{\mathrm \{sym\}^\{j\}f\}(n)\}\{n^\{s\}\}. \] When $j=2$, the error term which we obtain improves the earlier known result.},
author = {Sharma, Anubhav, Sankaranarayanan, Ayyadurai},
journal = {Czechoslovak Mathematical Journal},
keywords = {nonprincipal Dirichlet character; Hölder’s inequality; $j$th symmetric power $L$-function; holomorphic cusp form},
language = {eng},
number = {3},
pages = {885-901},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the average behavior of the Fourier coefficients of $j$th symmetric power $L$-function over certain sequences of positive integers},
url = {http://eudml.org/doc/299113},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Sharma, Anubhav
AU - Sankaranarayanan, Ayyadurai
TI - On the average behavior of the Fourier coefficients of $j$th symmetric power $L$-function over certain sequences of positive integers
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 885
EP - 901
AB - We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j \ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\rm sym}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb {Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum \[ S_j^*:= \sum _{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\le x (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in \mathbb {Z}^{6}} \lambda ^{2}_{{\rm sym}^{j}f}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}), \] where $x$ is sufficiently large, and \[ L(s,\mathrm {sym}^{j}f):=\sum _{n=1}^{\infty }\frac{\lambda _{\mathrm {sym}^{j}f}(n)}{n^{s}}. \] When $j=2$, the error term which we obtain improves the earlier known result.
LA - eng
KW - nonprincipal Dirichlet character; Hölder’s inequality; $j$th symmetric power $L$-function; holomorphic cusp form
UR - http://eudml.org/doc/299113
ER -