On the linear independence of the special values of a Dirichlet series with periodic coefficients

TL;DR

This paper generalizes T. Rivoal's result on the irrationality of special values of the Riemann zeta function to a broader class of Dirichlet series with periodic coefficients, providing lower bounds on their linear independence over rationals.

Contribution

It establishes a lower bound for the dimension of the rational vector space spanned by special values of Dirichlet series with periodic coefficients, extending previous results to more general series.

Findings

Special values at even and odd integers include infinitely many irrationals.

Provides a lower bound on the dimension of the space spanned by these values.

Generalizes Rivoal's result from the Riemann zeta function to broader Dirichlet series.

Abstract

A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers contain infinitely many irrational numbers. This result is proved by T.Rivoal if the function considered is the Riemann zeta function, and this paper gives its generalization to more general Dirichlet series.