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arXiv · 2307.01889

A computable formula for evaluating the mean square sum of $L$-functions

Neha Elizabeth Thomas·K Vishnu Namboothiri
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Abstract

For Dirichlet characters $\chi$ mod $k$ where $k\geq 3$, we here give a computable formula for evaluating the mean square sums $\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for any positive integer $r\geq 3$. We also give an inductive formula for computing the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\left(\sin\left(\frac{\pi m}{k}\right)\right)^{2n}}$ where $n$ is a positive integer in terms of Bernoulli numbers and binomial coefficients.

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Neha Elizabeth Thomas, K Vishnu Namboothiri. 2023-07-04. A computable formula for evaluating the mean square sum of $L$-functions. https://arxiv.org/abs/2307.01889

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