Irrationality of zeta values

Stéphane Fischler

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 27-62
  • ISSN: 0303-1179

Abstract

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The values of Riemann zeta function at positive even integers are transcendental numbers, since they are rational multiples of powers of π . On the contrary, very little is known about the arithmetic nature of ζ ( 2 k + 1 ) for positive integers k . Apéry proved in 1978 that ζ ( 3 ) is irrational. Rivoal proved in 2000 that infinitely many ζ ( 2 k + 1 ) are irrational, but without being able to construct any such k 2 . There are several ways to see Apéry’s proof; the one using hypergeometric series yields at the same time Apéry’s and Rivoal’s theorems.

How to cite

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Fischler, Stéphane. "Irrationalité de valeurs de zêta." Séminaire Bourbaki 45 (2002-2003): 27-62. <http://eudml.org/doc/252133>.

@article{Fischler2002-2003,
abstract = {Les valeurs aux entiers pairs (strictement positifs) de la fonction $\zeta $ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de $\pi $. En revanche, on sait très peu de choses sur la nature arithmétique des $\zeta (2k+1)$, pour $k \ge 1$ entier. Apéry a démontré en 1978 que $\zeta (3)$ est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de $\zeta (2k+1)$ sont irrationnels, mais sans pouvoir en exhiber aucun autre que $\zeta (3)$. Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques permet d’obtenir à la fois les théorèmes d’Apéry et de Rivoal.},
author = {Fischler, Stéphane},
journal = {Séminaire Bourbaki},
keywords = {irrationality; Riemann zeta function; hypergeometric series; Padé approximation; Apéry’s theorem; rational approximation; polylogarithm},
language = {fre},
pages = {27-62},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Irrationalité de valeurs de zêta},
url = {http://eudml.org/doc/252133},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Fischler, Stéphane
TI - Irrationalité de valeurs de zêta
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 27
EP - 62
AB - Les valeurs aux entiers pairs (strictement positifs) de la fonction $\zeta $ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de $\pi $. En revanche, on sait très peu de choses sur la nature arithmétique des $\zeta (2k+1)$, pour $k \ge 1$ entier. Apéry a démontré en 1978 que $\zeta (3)$ est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de $\zeta (2k+1)$ sont irrationnels, mais sans pouvoir en exhiber aucun autre que $\zeta (3)$. Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques permet d’obtenir à la fois les théorèmes d’Apéry et de Rivoal.
LA - fre
KW - irrationality; Riemann zeta function; hypergeometric series; Padé approximation; Apéry’s theorem; rational approximation; polylogarithm
UR - http://eudml.org/doc/252133
ER -

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