Arithmetic of linear forms involving odd zeta values

TL;DR

This paper introduces a hypergeometric method to construct linear forms in odd zeta values, recovering known irrationality measures and proving at least one of four specific zeta values is irrational.

Contribution

It presents a new hypergeometric construction that unifies and extends previous results on the irrationality of odd zeta values.

Findings

Reproduces Rhin and Viola's irrationality measures for ζ(2) and ζ(3)

Explains Rivoal's infinite-irrationality result

Proves at least one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational

Abstract

A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $\zeta(2)$ and $\zeta(3)$, as well as to explain Rivoal's "infinitely-many" result (math.NT/0008051) and to prove that at least one of the four numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, and $\zeta(11)$ is irrational.