Article
| Title: | The average behavior of coefficients of symmetric power $L$-functions over a certain sequence (English) |
| Author: | Wang, Pan |
| Author: | Wang, Tianze |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 1 |
| Year: | 2026 |
| Pages: | 1-16 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $f$ be a Hecke eigenform of even weight for the full modular group $SL(2,\mathbb {Z})$, and $L(s,{\rm sym}^{j} f)$, $j \ge 2,$ be the $j$th symmetric power $L$-function associated to $f$. Denote by $\lambda _{{\rm sym}^{j} f}(n)$ the $n$th normalized coefficient of the Dirichlet series of $L(s,{\rm sym}^{j} f)$. We study the average behavior of $\lambda _{{\rm sym}^{j} f}(n)$ and $\lambda _{{\rm sym}^{j} f}^{2}(n)$ over sums of squares of eight integers, i.e., $$\sum _{\substack {n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{8}^{2} \leq x \\(a_{1}, a_{2}, \cdots , a_{8}) \in \mathbb {Z}^{8}}}\lambda _{{\rm sym}^{j} f}(n)\quad \text {and}\quad \sum _{\substack {n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{8}^{2} \leq x \\(a_{1}, a_{2}, \cdots , a_{8}) \in \mathbb {Z}^{8}}}\lambda _{{\rm sym}^{j} f}^{2}(n),$$ and obtain the corresponding asymptotic formulas. (English) |
| Keyword: | Dirichlet coefficients |
| Keyword: | symmetric power $L$-function |
| Keyword: | average behavior |
| Keyword: | asymptotic formula |
| MSC: | 11F11 |
| MSC: | 11F30 |
| MSC: | 11F66 |
| DOI: | 10.21136/CMJ.2026.0446-24 |
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| Date available: | 2026-03-13T09:26:35Z |
| Last updated: | 2026-03-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153554 |
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