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

Abstract:

In this paper, we define $q$-analogues of Dirichlet’s beta function at positive integers, which can be written as $\beta _q(s)=\sum _{k\geq 1}\sum _{d|k}\chi (k/d)d^{s-1}q^k$ for $s\in \mathbb {N}^*$, where $q$ is a complex number such that $|q|<1$ and $\chi$ is the nontrivial 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 $\mathbb {Q}$ spanned by $1,\beta _q(2),\beta _q(4),\dots ,\beta _q(A)$, where $1/q\in \mathbb {Z}\setminus \{-1;1\}$ and $A$ is an even integer. As consequences for $1/q\in \mathbb {Z}\setminus \{-1;1\}$, on the one hand there is an infinity of irrational numbers among $\beta _q(2),\beta _q(4),\dots$, and on the other hand at least one of the numbers $\beta _q(2),\beta _q(4),\dots , \beta _q(20)$ is irrational.