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Basic hypergeometric series, $$q$$-analoges, of the values of the zeta function, and Eisenstein series. (Séries hypergéométriques basiques, $$q$$-analogues des valeurs de la fonction zêta et séries d’Eisenstein.) (French) Zbl 1089.11038
Following T. Rivoal [Acta Arith. 103, No. 2, 157–167 (2002; Zbl 1015.11033)] the authors prove the nice result that if $$q\in {\mathbb{C}} \setminus \{ 0, 1, -1\}$$ with $$\mid q\mid <1$$ and $$\zeta_q(s)=\sum_{k=1}^\infty q^k\sum_{d\mid k}d^{s-1}$$ for $$s=1,2,\dots$$ then the dimension of the vector space over $$\mathbb {Q}$$ which is spanned by $$1, \zeta_q(3),\zeta_q(5), \dots ,$$ $$\zeta_q(M)$$ where $$M$$ is sufficiently large odd integer is at least $$0.3358\sqrt{M}$$.

##### MSC:
 11J72 Irrationality; linear independence over a field 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
Zbl 1015.11033
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