Irrationalit\'e aux entiers impairs positifs d'un q-analogue de la fonction zeta de Riemann

TL;DR

This paper investigates a q-analogue of the Riemann zeta function at positive integers, establishing new lower bounds on the dimension of related vector spaces and demonstrating the irrationality of at least one among several specific values.

Contribution

It provides an improved lower bound on the dimension of the space spanned by q-analogue zeta values, advancing understanding of their algebraic independence.

Findings

New lower bound for the dimension of spanned vector space

At least one of the specified q-analogue zeta values is irrational

Improves previous results by Krattenthaler, Rivoal, and Zudilin

Abstract

In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s\in\N^* by \zeta_q(s)=\sum_{k\geq 1}q^k\sum_{d|k}d^{s-1}. We give a new lower bound for the dimension of the vector space over \Q spanned, for 1/q\in\Z\setminus\{-1;1\} and an even integer A, by 1,\zeta_q(3),\zeta_q(5),...,\zeta_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin (\emph{S\'eries hyperg\'eom\'etriques basiques, q-analogues des valeurs de la fonction zeta et s\'eries d'Eisenstein}, J. Inst. Jussieu {\bf 5}.1 (2006), 53-79). In particular, a consequence of our result is that for 1/q\in\Z\setminus\{-1;1\}, at least one of the numbers \zeta_q(3),\zeta_q(5),\zeta_q(7),\zeta_q(9) is irrational.