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arXiv · 2512.02612

Linear independence of values of Dirichlet $L$ functions

Ludovic Mistiaen
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Abstract

In this paper, for a given Dirichlet character mod $N$ with $4\nmid N$, we give a lower bound of order $\sqrt{s/\log(s)}$ for the dimension of the $\mathbb{Q}(e^{2i\pi/N})$-vector space spanned by the values of its $L$-function at integers $\leq s$ of a given parity. We thus generalize a result Fischler proved in 2021, corresponding to the principal character mod 1. To this end, we construct linear combinations of these values of $L$-function with a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on a ``Shidlovskii's lemma''.

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Ludovic Mistiaen. 2025-12-02. Linear independence of values of Dirichlet $L$ functions. https://arxiv.org/abs/2512.02612

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