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

Abstract

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 $\zeta (2)$ 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 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 $\mu ({\zeta }_q(2))<3.518876$ 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. 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.

118 pages; in Russian

References in corpus (5)

• Ramanujan-type supercongruences
• On the Mahler measure of $1+X+1/X+Y+1/Y$
• Apery limits and special values of L-functions
• Linear Mahler Measures and Double L-values of Modular Forms
• Ramanujan-type formulae for $1/\pi$: the art of translation