Hypergeometric series and irrationality of the values of the Riemann zeta function
type: Item
instance of: scholarly article
title: Hypergeometric series and irrationality of the values of the Riemann zeta function
zbMATH Open document ID: 1041.11051
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/223040
review text: The author presents a survey of the ideas around his striking discoveries about the values of the Riemann zeta-function at the odd integers. He shows how hypergeometric series and Padé approximants provide a common approach to Apéry's theorem on the irrationality of \(\zeta(3)\), the author's remarkable proof that infinitely many of the \(\zeta(2n+1)\) are irrational, and Zudilin's theorem, improving another result of the author, that at least one of the numbers \(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\) is irrational. Some teasing questions are added in the conclusion.
Mathematics Subject Classification ID: 11J72
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 2058873
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: hypergeometric series
zbMATH Keywords: Padé approximants
author: T. Rivoal
reviewed by: John H. Loxton
MaRDI profile type: Publication
OpenAlex ID: W2014025499
cites work: Q4184941
cites work: Q3912668
cites work: Irrationality of infinitely many values of the zeta function at odd integers.
cites work: A Note on the Irrationality of ζ(2) and ζ(3)
cites work: Q3335852
cites work: Q3934475
cites work: On the method of Thue-Siegel
cites work: Q5736202
cites work: Q3961696
cites work: Padé approximants and balanced hypergeometric series.
cites work: THE IRRATIONALITY OF CERTAIN QUANTITIES INVOLVING ζ(3)
cites work: Legendre type polynomials and irrationality measures.
cites work: On the linear independence of numbers
cites work: A few remarks on \(\zeta (3)\)
cites work: ON IRRATIONALITY OF THE VALUES OF THE FUNCTIONS $ F(x,s)$
cites work: Hypergeometric functions
cites work: A new proof of the irrationality of \(\zeta (2)\) and \(\zeta (3)\) using Padé approximants
cites work: On a permutation group related to ζ(2)
cites work: The group structure for \(\zeta(3)\)
cites work: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs
cites work: Irrationalité d'au moins un des neuf nombres ζ(5), ζ(7),...,ζ(21)
cites work: Diophantine properties of numbers related to Catalan's constant
cites work: Q3773958
cites work: Q5508372
cites work: Apéry's theorem
cites work: A proof that Euler missed. Apéry's proof of the irrationality of \(\zeta(3)\). An informal report
cites work: Multiple zeta values: an introduction
DOI: 10.5802/JTNB.406
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description: scientific article; zbMATH DE number 2058873