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Keywords:
nonprincipal Dirichlet character; Hölder's inequality; $j$th symmetric power $L$-function; holomorphic cusp form
nonprincipal Dirichlet character; Hölder's inequality; $j$th symmetric power $L$-function; holomorphic cusp form
Summary:
We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j \geq 2$ 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 $$ S_j^*:= \sum_{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq x (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in \mathbb {Z}^{6}} \lambda ^{2}_{{\rm sym}^{j}f}(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}), $$ 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.
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