Article
| Title: | Mean values related to the Dedekind zeta-function (English) |
| Author: | Tang, Hengcai |
| Author: | Wang, Youjun |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 4 |
| Year: | 2024 |
| Pages: | 1265-1274 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $K/\mathbb {Q}$ be a nonnormal cubic extension which is given by an irreducible polynomial $g(x)=x^3+a x^2+b x+c$. Denote by $\zeta _{K}(s)$ the Dedekind zeta-function of the field $K$ and $a_K(n)$ the number of integral ideals in $K$ with norm $n$. In this note, by the higher integral mean values and subconvexity bound of automorphic $L$-functions, the second and third moment of $a_K(n)$ is considered, i.e., $$ \sum _{n\leq x}a_K^2(n)=x P_1(\log x)+O(x^{5/7+\epsilon }),\quad \sum _{n\leq x}a_K^3(n)=x P_4(\log x)+O(X^{321/356+\epsilon }), $$ where $P_1(t)$, $P_4(t)$ are polynomials of degree 1, 4, respectively, $\epsilon >0$ is an arbitrarily small number. (English) |
| Keyword: | cusp form |
| Keyword: | Dedekind zeta-function |
| Keyword: | $L$-function |
| MSC: | 11F11 |
| MSC: | 11F30 |
| MSC: | 11F66 |
| MSC: | 11N37 |
| DOI: | 10.21136/CMJ.2024.0252-24 |
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| Date available: | 2024-12-15T06:41:26Z |
| Last updated: | 2024-12-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152700 |
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| Reference: | [9] Lin, Y., Nunes, R., Qi, Z.: Strong subconvexity for self-dual GL(3) $L$-functions.Int. Math. Res. Not. 2023 (2023), 11453-11470. Zbl 1541.11046, MR 4609788, 10.1093/imrn/rnac153 |
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