Apéry's theorem and problems for the values of Riemann's zeta function and their q-analogues

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This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of ζ(2) and ζ(3), the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers ζ(5), ζ(7), ζ(9) and ζ(11) is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the generalized multiple integrals introduced by Beukers in his proof of Apéry's result and the very-well-poised hypergeometric series; it is based on arXiv:math.CA/0206177. Chapter 3 surveys some arithmetic and hypergeometric q-analogies and establishes the irrationality measure μ(ζq(2))<3.518876 for a q-analogue of ζ(2); it closely follows the text in Sb. Math. 193 (2002), 1151--1172, but also incorporates the sharper analysis of the hypergeometric construction by Smet and Van Assche (arXiv:0809.2501 [math.CA]) to produce the improvement upon the 2002 result. Chapter 4 is devoted to the measure μ(ζ(2))<5.095412 and is based on arXiv:1310.1526 [math.NT]; Chapter 5 is establishing the estimate ||(3/2)ᵏ||>0.5803ᵏ for the distance from (3/2)ᵏ to the nearest integer, with the English version published in J. Théor. Nombres Bordeaux 19 (2007), 313--325. Chapter 6 reproduces the solution (from arXiv:math.CA/0311195) to the problem of Asmus Schmidt about generalized Apéry's numbers. Finally, Chapter 7 is about expressing the special L-values as periods (in the sense of Kontsevich and Zagier), in particular, as values of hypergeometric functions; it is based on the publication in Springer Proc. Math. Stat. 43 (2013), 381--395.