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

2007/12/11 by Frederic Jouhet, Jouhet, Frederic, Elie Mosaki +1
Mathematics · #11J72 (Primary) #33D15 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11J72 #msc:33D15

paper · pdf · doi:10.48550/arxiv.0712.1762

33 pages

arxiv created 2007/12/11 · arxiv updated 2009/12/01

Abstract

In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s∈\N^* by ζ_q(s)=∑_{k≥ 1}q^k∑_d|kd^s-1. We give a new lower bound for the dimension of the vector space over \Q spanned, for 1/q∈\Z∖\-1;1\ and an even integer A, by 1,ζ_q(3),ζ_q(5),...,ζ_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin (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∈\Z∖\-1;1\, at least one of the numbers ζ_q(3),ζ_q(5),ζ_q(7),ζ_q(9) is irrational.

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• La Zeta de Riemann est irrationnelle aux impairs positifs
• Infinitely many odd zeta values are irrational. By elementary means