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

TL;DR

This paper introduces q-analogues of Dirichlet's beta function at positive integers, explores their algebraic properties, and proves the transcendence and irrationality of certain values, linking number theory with automorphic forms.

Contribution

It defines new q-analogues of Dirichlet's beta function, establishes their connection with automorphic forms, and proves transcendence and irrationality results for their values.

Findings

Transcendence of eta_q(2s+1) for algebraic q with |q|<1

Lower bounds on the dimension of rational span of eta_q(2), eta_q(4), ..., eta_q(A)

Infinitely many irrational values among eta_q(2), eta_q(4), ...

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.