Irrationalit\'e de valeurs de z\^eta (d'apr\`es Ap\'ery, 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 Ap\'ery'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.