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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