Linear independence of values of Dirichlet $L$ functions

TL;DR

This paper establishes a lower bound on the dimension of the space spanned by values of Dirichlet L-functions at integers, extending previous results and employing advanced linear independence criteria and lemmas.

Contribution

It generalizes Fischler's 2021 result to non-principal characters and introduces refined techniques for analyzing linear independence of L-function values.

Findings

Lower bound of order √(s/log s) for the dimension

Extension of Fischler's result to non-principal characters

Application of a generalized linear independence criterion

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''.