Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zêta et séries d’Eisenstein

We study the arithmetic properties of q-analogues of values ζ(s) of the Riemann zeta function, in particular of the values of the functions ζq(s)=Σ∞ k=₁ qkΣ dk d s⁻¹, s= 1,2,..., where q is a complex number with |q|<1 (these functions are also connected with the automorphic world). The main theorem of this article is that, if 1/q is an integer different from ±1, and if M is a sufficiently large odd integer, then the dimension of the vector space over ℚ which is spanned by 1,ζq(3),ζq(5),...,ζq(M) is at least c₁√M}, where c₁=0.3358. This result can be regarded as a q-analogue of the result of Rivoal and of Ball and Rivoal that the dimension of the vector space over ℚ which is spanned by 1,ζ(3),ζ(5),...,ζ(M) is at least c₂log M, with c₂=0.5906. For the same values of q, a similar lower bound for the values ζq(s) at even integers s provides a new proof of a special case of a result of Bertrand saying that one of the two Eisenstein series E₄(q) and E₆(q) is transcendental over ℚ for any complex number q such that 0<|q|<1.