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arXiv · 2405.17981

Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers

Stéphane Louboutin
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Abstract

Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{\phi(f)}\sum_{\chi (-1)=(-1)^m}\vert L(m,\chi )\vert^2,$$ where $\chi$ ranges over the $\phi (f)/2$ 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 $L(m_1,\chi_1)\cdots L(m_{n-1},\chi_{n-1})\overline{L(m_n,\chi_1\cdots\chi_{n-1})}$.

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Stéphane Louboutin. 2024-05-28. Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers. https://arxiv.org/abs/2405.17981

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