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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 \(0.3358\sqrt{M}\).

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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