This material has been published in J. Inst. Math. Jussieu 5 (2006), 53-79, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Cambridge University Press. This material may not be copied or reposted without explicit permission.

Christian Krattenthaler, Tanguy Rivoal and Wadim Zudilin

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

(26 pages)

English Abstract. We study the arithmetic properties of certain q-analogues of values \zeta(j) of the Riemann zeta function, in particular of the values of the functions \zetaq(s)= \sum _{k=1} ^{\infty}qk \sum _{d|k}ds-1, s=1,2,..., where q is a complex number with |q|<1. The main theorem of this article is that, if 1/q is an integer different from -1 and 1, and if M is a sufficiently large odd integer, then the dimension of the vector space over Q which is spanned by 1,\zetaq(3), \zetaq(5), ..., \zetaq(M) is at least c1M1/2, where c1=0,3358. This result can be regarded as a q-analogue of the result [14,2] that the dimension of the vector space over Q which is spanned by 1,\zeta(3), \zeta(5), ..., \zeta(M) is at least c2log M, with c2=0,5906. For the same values of q, a similar lower bound for the values \zeta1(s) at even integers s provides a new proof of a special case of a result of Bertrand [Bull. Soc. Math. France 104 (1976), 309-321] 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.

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