Basic hypergeometric series, q-analogues of the values of the function zeta and the eisenstein series

We study the arithmetic properties of q-analogues of values zeta(s) of the Riemann zeta function, in particular of the values of the functions zeta(q)(s) = Sigma(k=1 q)(infinity)(k) Sigma(d vertical bar k) d(s-1), s=1, 2,..., where q is a number with vertical bar q vertical bar < 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, zeta(q)(3), zeta(q)(5),..., zeta(q)(M) is at least c(1) root M, where c(1) = 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, zeta(3), zeta(5),..., zeta(M) is at least c(2) log M, with c(2) = 0.5906. For the same values of q, a similar lower bound for the values zeta(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 E-4 (q) and E-6 (q) is transcendental over Q for any complex number q such that 0 < vertical bar q vertical bar < 1.