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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.17228 |
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| _version_ | 1866908425648078848 |
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| author | Hong, Ziwei Fang, Zhongqiu |
| author_facet | Hong, Ziwei Fang, Zhongqiu |
| contents | We investigate the mean value of the first moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum
\begin{equation*}
\sum_{\substack{χ primitive\ cubic\\ genus(χ)=g}}L_q(\frac{1}{2}, χ),
\end{equation*} where $L_q(s,χ)$ denotes the $L$-function associated with primitive cubic character $χ$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{7}{8}+\varepsilon)g}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean value of cubic $L$-funcitons with fixed genus Hong, Ziwei Fang, Zhongqiu Number Theory We investigate the mean value of the first moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ cubic\\ genus(χ)=g}}L_q(\frac{1}{2}, χ), \end{equation*} where $L_q(s,χ)$ denotes the $L$-function associated with primitive cubic character $χ$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{7}{8}+\varepsilon)g}$. |
| title | Mean value of cubic $L$-funcitons with fixed genus |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.17228 |