  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 Ap\'ery's proof from 1978 of the
irrationality of $\zeta(2)$ and $\zeta(3)$, the values of Riemann's zeta
function. Chapter 1 is about "at least one of the four numbers $\zeta(5)$,
$\zeta(7)$, $\zeta(9)$ and $\zeta(11)$ 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 Ap\'ery'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
$\mu(\zeta_q(2))<3.518876$ for a $q$-analogue of $\zeta(2)$; 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 $\mu(\zeta(2))<5.095412$ and is based on
arXiv:1310.1526 [math.NT]; Chapter 5 is establishing the estimate
$||(3/2)^k||>0.5803^k$ for the distance from $(3/2)^k$ to the nearest integer,
with the English version published in J. Th\'eor. Nombres Bordeaux 19 (2007),
313--325. Chapter 6 reproduces the solution (from arXiv:math.CA/0311195) to the
problem of Asmus Schmidt about generalized Ap\'ery'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.

Number Theory;Algebraic Geometry;Classical Analysis and ODEs;Combinatorics