Irrationalité de valeurs de zêta  [ Irrationality of zeta values ]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 910, p. 27-62

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(2k+1) for positive integers k. Apéry proved in 1978 that ζ(3) is irrational. Rivoal proved in 2000 that infinitely many ζ(2k+1) are irrational, but without being able to construct any such k2. 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.

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(2k+1), pour k1 entier. Apéry a démontré en 1978 que ζ(3) est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de ζ(2k+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.

Classification:  11J72,  11G55,  11M06,  33C20,  41A21
Keywords: irrationality, Riemann zeta function, hypergeometric series, Padé approximation, Apéry's theorem, rational approximation, polylogarithm
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     author = {Fischler, St\'ephane},
     title = {Irrationalit\'e de valeurs de z\^eta},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
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     series = {Ast\'erisque},
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Fischler, Stéphane. Irrationalité de valeurs de zêta, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 910, pp. 27-62. http://www.numdam.org/item/SB_2002-2003__45__27_0/

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