Irrationalité aux entiers impairs positifs d'un q-analogue de la fonction zeta de Riemann

Frederic Jouhet (1) , Elie Mosaki (1)

1 ICJ - Institut Camille Jordan

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),\dots,\zeta_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin (\emph{Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zeta et séries 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.

Mots clés: q-analogue of the Riemann zeta function; irrationality; basic hypergeometric series

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Frederic Jouhet, Elie Mosaki. Irrationalité aux entiers impairs positifs d'un q-analogue de la fonction zeta de Riemann. International Journal of Number Theory, 2010, 6, pp.959-988. ⟨hal-00195936⟩