DIOPHANTINE PROBLEMS FOR q-ZETA VALUES 1

hold. The above defined q-zeta values (1) present several new interesting problems in the theory of diophantine approximations and transcendental numbers; these problems are extensions of the corresponding problems for ordinary zeta values and we state some of them in Section 3 of this note. Our nearest aim is to demonstrate how some recent contributions to the arithmetic study of the numbers ζ(k), k = 2, 3, . . . , successfully work for q-zeta values. Namely, we mean the hypergeometric construction of linear forms (proposed in the works of E.M. Nikishin [2], L.A. Gutnik [3], Yu.V. Nesterenko [4]) and the arithmetic method (due to G.V. Chudnovsky [5], E.A. Rukhadze [6], M. Hata [7]) accompanied with the group-structure scheme (due to G. Rhin and C. Viola [8], [9]). The next section contains new irrationality measures of the numbers ζq(1) and ζq(2) for q −1 = p ∈ Z\{0,±1}, and our starting point is the following table illustrating a connection of some objects and their q-extensions (here ⌊ · ⌋ denotes the integral part of a number and the notation ‘l.c.m.’ means the least common multiple). We refer the reader to the book [10] and the works [11]–[13], where a motivation and a ground are presented.