- Title
- Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zêta et séries d’Eisenstein
- Creator
- Krattenthaler, C.; Rivoal, T.; Zudilin, W.
- Relation
- Journal of the Institute of Mathematics of Jussieu Vol. 5, Issue 1, p. 53-79
- Publisher Link
- http://dx.doi.org/10.1017/S1474748005000149
- Publisher
- Cambridge University Press
- Resource Type
- journal article
- Date
- 2006
- Description
- We study the arithmetic properties of q-analogues of values ζ(s) of the Riemann zeta function, in particular of the values of the functions ζq(s)=Σ∞ k=₁ qkΣ dk d s⁻¹, s= 1,2,..., where q is a complex number with |q|<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,ζq(3),ζq(5),...,ζq(M) is at least c₁√M}, where c₁=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,ζ(3),ζ(5),...,ζ(M) is at least c₂log M, with c₂=0.5906. For the same values of q, a similar lower bound for the values ζq(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 E₄(q) and E₆(q) is transcendental over ℚ for any complex number q such that 0<|q|<1.
- Subject
- q-analogues of the Riemann zeta function; modular forms; Eisenstein series; basic hypergeometric series
- Identifier
- uon:6474
- Identifier
- http://hdl.handle.net/1959.13/803678
- Identifier
- ISSN:1474-7480
- Rights
- © Cambridge University Press. Reprinted with permission.
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