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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.17981 |
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| _version_ | 1866914814717067264 |
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| author | Louboutin, Stéphane |
| author_facet | Louboutin, Stéphane |
| contents | Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{ϕ(f)}\sum_{χ(-1)=(-1)^m}\vert L(m,χ)\vert^2,$$ where $χ$ ranges over the $ϕ(f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,χ_1)\cdots L(m_{n-1},χ_{n-1})\overline{L(m_n,χ_1\cdotsχ_{n-1})}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers Louboutin, Stéphane Number Theory Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{ϕ(f)}\sum_{χ(-1)=(-1)^m}\vert L(m,χ)\vert^2,$$ where $χ$ ranges over the $ϕ(f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,χ_1)\cdots L(m_{n-1},χ_{n-1})\overline{L(m_n,χ_1\cdotsχ_{n-1})}$. |
| title | Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.17981 |