Article
| Title: | On the average behavior of the Fourier coefficients of $j$th symmetric power $L$-function over certain sequences of positive integers (English) |
| Author: | Sharma, Anubhav |
| Author: | Sankaranarayanan, Ayyadurai |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 885-901 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | nonprincipal Dirichlet character |
| Keyword: | Hölder's inequality |
| Keyword: | $j$th symmetric power $L$-function |
| Keyword: | holomorphic cusp form |
| MSC: | 11F11 |
| MSC: | 11F30 |
| MSC: | 11M06 |
| idZBL: | Zbl 07729543 |
| idMR: | MR4632863 |
| DOI: | 10.21136/CMJ.2023.0348-22 |
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| Date available: | 2023-08-11T14:27:29Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151780 |
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