Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...)

Abstract

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Apéry's Theorem (1978) : $\zeta (3)$ is irrational. The second section is devoted to a variant of the proof, published by Rivoal and Ball-Rivoal, that infinitely many $\zeta (2n+1)$ are irrational. The end of this text deals with more quantitative statements.

Bourbaki Seminar, November 2002 ; to appear in Astérisque ; 45 pages ; in french

Cited by in corpus (7)

• A method for determining the mod-$p^k$ behaviour of recursive sequences
• On a linear form for Catalan's constant
• From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve
• Open Diophantine Problems
• Auxiliary functions in transcendence proofs
• A simplification of Apéry's proof of the irrationality of ζ(3)
• Séries hypergéométriques basiques, $q$-analogues des valeurs de la fonction zêta et séries d'Eisenstein