Article
| Title: | A note on average behaviour of the Fourier coefficients of $j$\lowercase {th} symmetric power $L$-function over certain sparse sequence of positive integers (English) |
| Author: | Wang, Youjun |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 2 |
| Year: | 2024 |
| Pages: | 623-636 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $j\geq 2$ be a given integer. Let $H_{k}^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ 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 $$ \sum _{n=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq x \atop (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in \mathbb {Z}^{ 6}} \lambda _{{\rm sym}^{j}f}^{2}(n), $$ where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023). (English) |
| Keyword: | cusp form |
| Keyword: | Fourier coefficient |
| Keyword: | symmetric power $L$-function |
| MSC: | 11F11 |
| MSC: | 11F30 |
| MSC: | 11F66 |
| DOI: | 10.21136/CMJ.2024.0038-24 |
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| Date available: | 2024-07-10T14:59:13Z |
| Last updated: | 2024-07-15 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152462 |
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