Linear independence of values of Dirichlet $L$ functions
Ludovic Mistiaen
TL;DR
The paper proves a quantitative lower bound on the dimension, over the cyclotomic field K = Q(e^{2πi/N}), of the space spanned by Dirichlet L-values L(χ,i) at integers i of a fixed parity up to s, for any odd N with N ≠ 0 mod 4 and χ primitive modulo N. It develops non-explicit rational functions F_n and uses refined Siegel-type diophantine technology together with a Shidlovskii-type zero lemma for a differential system tied to polylogarithms, to construct linear forms Λ_{n,(p,k)} that become linearly independent in the limit. The core novelty lies in coupling Fischler’s non-explicit combination method with a Shidlovskii-amenable framework to achieve a universal lower bound ~0.42 N^{−3/2} sqrt(s/log s) for the dimension over the cyclotomic field, extending previous zeta-value results to Dirichlet L-values. Fourier/inversion techniques and Gauss sums filter terms by parity, and the approach recovers known outcomes when N=1 while remaining nontrivial for general N.
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 $\mathbb{Q}(e^{2iπ/N})$-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''.