PreprintCC BY-ND 4.0HAL
Linear independence of values of Dirichlet L functions
Abstract
In this paper, for a given Dirichlet character mod N with 4nmid N, we give a lower bound of order √(s/log(s)) for the dimension of the ℚ(e2iπ/N)-vector space spanned by the values of its L-function at integers ≤ 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".
Related
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Extreme values of Dirichlet L-functions on the critical line with a growing prime modulus
- A linear independence criterion for p-adic Hurwitz zeta values at odd integers
- Irrationality of values of L-functions for real Dirichlet characters
- The threshold for linear independence of multiple zeta values in positive characteristic
BibTeX
@misc{mistiaen2026linear,
title = {{Linear independence of values of Dirichlet $L$ functions}},
author = {Ludovic Mistiaen},
howpublished = {HAL},
year = {2026},
month = sep,
eprint = {2512.02612},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05390882v2},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - Linear independence of values of Dirichlet L functions
AU - Ludovic Mistiaen
PY - 2026
DA - 2026/09/27
PB - HAL
UR - https://hal.science/hal-05390882v2
AB - In this paper, for a given Dirichlet character mod N with 4nmid N, we give a lower bound of order √(s/log(s)) for the dimension of the ℚ(e^2iπ/N)-vector space spanned by the values of its L-function at integers ≤ 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".
ER -