Irrationality of Values of Zeta-function

1. Introduction. The irrationality of values of the zeta-function ζ(s) at odd integers s ≥ 3 is one of the most attractive problems in number theory. Inspite of a deceptive simplicity and more than two-hundred-year history of the problem, all done in this direction can easily be counted. It was only 1978, when Apéry [A] obtained the irrationality of ζ(3) by a presentation of " nice " rational approximations to this number. During next years the phenomenon of Apéry's sequence was recomprehended more than once from positions of different analytic methods (see [N2] and the bibliography cited there); new approaches gave rise to improve Apéry's result quantitatively, i.e., to get a " sharp " irrationality measure of ζ(3) (last stages in this direction are the articles [H2], [RV]). Finally, in 2000 Rivoal [R1] constructed linear forms with rational coefficients involving values of ζ(s) only at odd integers s > 1 and proved that there exist infinitely many irrational numbers among ζ(3), ζ(5), ζ(7),. .. ; more precisely, for the dimension δ(a) of spaces spanned over Q by 1, ζ(3), ζ(5),. .. , ζ(a − 2), ζ(a), where a is odd , there holds the estimate