A Liouville-type estimate is proved for the irrationality measure of the quantities n=1; q; n; (1 q n; 2q (2) = (1 q n with q 2 Z n f0; 1g. The proof is based on the application of a q-analogue of the arithmetic method developed by Chudnovsky, Rukhadze, and Hata and of the transformation group for hypergeometric series -- the group-structure approach introduced by Rhin and Viola.

this paper, we denote by ord '(z) the multiplicity of zero of the function '(z) at the point z = 0. Here are some remarks concerning the structure of this paper. The facts which are presented in Section 1, originate in the general elimination theory. The content of Section 1 consists in an adaptation of the results of this theory to the needs of the theory of transcendental numbers. The corresponding technical results for the polynomial rings Z[x 1 ; : : : ; xm ] and C [z; x 1 ; : : : ; xm ] were described by the author in [10-15]. A dierent version of presentation can be found in the paper [16] by Philippon. The aim of Section 1 is the unied presentation of the algebraic foundations of the method, which allow one to apply the obtained technical assertions in the case of integer (Theorem 1) and algebraic (Theorem 2) coecients of polynomials as well as in the case of coecients from the eld C (z) (Theorem 3). In Section 2, we present an algebraic construction of polynomials of the values of Ramanujan functions, which generalizes the corresponding result of [3]. These polynomials are used in Section 3 to prove Theorem 2 and in Section 4 to prove Theorem 1.

The closedness of the system of thetanulls (and the Siegel modular forms) and their first derivatives with respect to differentiation is well-known in the one-dimensional case. It is shown in the present paper that thetanulls and their various logarithmic derivatives satisfy a non-linear system of differential equations; only one and two-dimensional versions of this result were known before. Several distinct examples of such systems are presented, and a theorem on the transcendence degree of the differential closure of the eld generated by all thetanulls is established. On the basis of a study of the modular properties of logarithmic derivatives of thetanulls (previously unknown) relations between these functions and thetanulls themselves are obtained in dimensions 2 and 3.

Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler’s Ɣ-function. Let q be a positive integer greater than 1 and γ denote Euler’s constant. We show that all the numbers ψ(a/q)+ γ, (a,q) = 1, 1 � a � q, are transcendental. We also prove that at most one of the numbers is algebraic.

doi:10.1112/blms/bdr031

Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt Transcendence of the log gamma function and some discrete