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

TL;DR

This survey reviews proofs of the irrationality and linear independence of values of the Riemann zeta function at positive odd integers, highlighting key results and recent quantitative advances.

Contribution

It compiles all known proofs of Apéry's theorem and discusses recent variants and quantitative results on the irrationality of zeta values at odd integers.

Findings

All known proofs of Apéry's theorem are presented.

Rivoal's and Ball-Rivoal's variants show infinitely many zeta(2n+1) are irrational.

Recent quantitative statements about zeta values are discussed.

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.