Linear independence of values of Dirichlet L functions

2025/12/02 by Mistiaen, Ludovic
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.02612

openalex publication_date 2025/12/02 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

In this paper, for a given Dirichlet character mod N with 4\nmid 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''.

Citations

• A note on the number of irrational odd zeta values, II
• Small improvements on the Ball-Rivoal theorem and its p-adic variant
• Linear independence of odd zeta values using Siegel's lemma
• Shidlovsky's multiplicity estimate and Irrationality of zeta values
• On the linear independence of the special values of a Dirichlet series with periodic coefficients
• An identity involving the least common multiple of binomial coefficients and its application

Related

• Explicit Subconvexity Estimates for Dirichlet L-functions
• Estimates for character sums and Dirichlet L-functions to smooth moduli
• An explicit upper bound for L(1, χ) when χ is quadratic
• Refined Upper Bounds for L(1,χ)
• Upper bounds for |L(1,chi)|