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

Krattenthaler, C.; Rivoal, T.; Zudilin, W.

J. Inst. Math. Jussieu 5, No. 1, 53-79 (2006).

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 \(0.3358\sqrt{M}\).

Reviewer: Jaroslav Hančl (Ostrava)

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, \({}_r\phi_s\)

Keywords:

Riemann zeta function; linear independence; dimension of vector space

Citations:

Zbl 1015.11033