Irrationalité de valeurs de zêta

Fischler, Stéphane

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 910, pp. 27-62.

Résumé
Abstract

Les valeurs aux entiers pairs (strictement positifs) de la fonction $ζ$ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de $π$ . En revanche, on sait très peu de choses sur la nature arithmétique des $ζ (2 k + 1)$ , pour $k \geq 1$ entier. Apéry a démontré en 1978 que $ζ (3)$ est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de $ζ (2 k + 1)$ sont irrationnels, mais sans pouvoir en exhiber aucun autre que $ζ (3)$ . 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.

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 $k \geq 2$ . 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.

EuDML MR Zbl | 3 citations dans Numdam

Classification : 11J72, 11G55, 11M06, 33C20, 41A21
Mot clés : irrationalité, fonction zêta de Riemann, série hypergéométrique, approximant de Padé, théorème d'Apéry, approximation rationnelle, polylogarithme
Keywords: irrationality, Riemann zeta function, hypergeometric series, Padé approximation, Apéry's theorem, rational approximation, polylogarithm

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Fischler, Stéphane. Irrationalité de valeurs de zêta, dans Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 910, pp. 27-62. http://www.numdam.org/item/SB_2002-2003__45__27_0/

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