# Two hypergeometric tales and a new irrationality measure of $$\zeta (2)$$ζ(2)

@article{Zudilin2013TwoHT,
title={Two hypergeometric tales and a new irrationality measure of \$\$\zeta (2)\$\$$\zeta(2)}, author={W. Zudilin}, journal={Annales math{\'e}matiques du Qu{\'e}bec}, year={2013}, volume={38}, pages={101-117} } • W. Zudilin • Published 2013 • Mathematics • Annales mathématiques du Québec We prove the new upper bound$$5.095412$$5.095412 for the irrationality exponent of$$\zeta (2)=\pi ^2/6$$ζ(2)=π2/6; the earlier record bound$$5.441243$$5.441243 was established in 1996 by G. Rhin and C. Viola.RésuméNous obtenons une nouvelle borne pour l’exposant d’irrationnalité de$$\zeta (2)=\pi ^2/6$$ζ(2)=π2/6, à savoir$$5.095412$$5.095412, cette dernière améliorant le record$$5.441243$$5.441243 établi par G. Rhin et C. Viola. On simultaneous diophantine approximations to$\zeta(2)$and$\zeta(3)$• Mathematics • 2013 The authors present a hypergeometric construction of rational approximations to$\zeta(2)$and$\zeta(3)$which allows one to demonstrate simultaneously the irrationality of each of the zeta values,Expand Irrationality proofs for zeta values, moduli spaces and dinner parties A simple geometric construction on the moduli spaces$\mathcal{M}_{0,n}$of curves of genus$0$with$n$ordered marked points is described which gives a common framework for many irrationalityExpand On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums • Mathematics, Physics • 2018 In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier–Legendre series expansions,Expand Do algebraic numbers follow Khinchin’s Law?∗ This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree > 2 separately. For random numbers theExpand Hypergeometric rational approximations to ζ(4) • Mathematics • Proceedings of the Edinburgh Mathematical Society • 2020 Abstract We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novelExpand Hypergeometric heritage of W. N. Bailey We review some of W.N. Bailey's work on hypergeometric functions that found solid applications in number theory. The text is complemented by Bailey's letters to Freeman Dyson from the 1940s. Vectors of type II Hermite–Padé approximations and a new linear independence criterion We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values ofExpand #### References SHOWING 1-10 OF 24 REFERENCES The group structure for ζ(3) • Mathematics • 2001 1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. ..Expand Ramanujan-type formulae and irrationality measures of some multiples of$ {\pi}\$
An explicit construction of simultaneous Pade approximations for gener- alized hypergeometric series and formulae for the quantities π √ d , d ∈{ 1, 2, 3, 10005}, in terms of these series are usedExpand
Well-poised hypergeometric service for diophantine problems of zeta values | NOVA. The University of Newcastle's Digital Repository
On montre comment les concepts classiques de series et integrales hypergeometriques bien equilibrees devient crucial dans l'etude des proprietes arithmetiques des valeurs de la fonction zeta deExpand
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The story exposed in this paper starts in 1978, when R. Apery [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apery’sExpand
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This also gives in the paper T. H. Koornwinder, Orthogonal polynomials with weight function (1− x)α(1 + x)β + Mδ(x + 1) + Nδ(x− 1), Canad. Math. Bull. 27 (1984), 205–214 the identitity (2.5) with N =Expand
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A new proof of the irrationality of the number ζ(3) is proposed. A new decomposition of this number into a continued fraction is found. Recurrence relations are proved for some sequences ofExpand
Арифметические гипергеометрические ряды@@@Arithmetic hypergeometric series
• Mathematics
• 2011
The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems. OriginallyExpand