Diophantine properties for q-analogues of Dirichlet's beta function at positive integers

Abstract

small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $\beta_q(s)=\sum_{k\geq1}\sum_{d|k}\chi(k/d)d^{s-1}q^k$ for $s\in\N^*$, where $q$ is a complex number such that $|q|<1$ and $\chi$ is the non trivial Dirichlet character modulo 4. For odd $s$, these expressions are connected with the automorphic world, in particular with Eisenstein series of level 4. From this, we derive through Nesterenko's work the transcendance of the numbers $\beta_q(2s+1)$ for $q$ algebraic such that $0<|q|<1$. Our main result concerns the nature of the numbers $\beta_q(2s)$: we give a lower bound for the dimension of the vector space over $\Q$ spanned by $1,\beta_q(2),\beta_q(4),...,\beta_q(A)$, where $1/q\in\Z\setminus\{-1;1\}$ and $A$ is an even integer. As consequences, for $1/q\in\Z\setminus\{-1;1\}$, on the one hand there is an infinity of irrational numbers among $\beta_q(2),\beta_q(4),...$, and on the other hand at least one of the numbers $\beta_q(2),\beta_q(4),..., \beta_q(20)$ is irrational.