Infinitely many odd zeta values are irrational. By elementary means

TL;DR

This paper presents an elementary proof demonstrating that infinitely many odd zeta values are irrational, simplifying previous complex proofs that relied on advanced mathematical techniques.

Contribution

It provides the first elementary proof of the infinitude of irrational odd zeta values, extending Zudilin's approach with hypergeometric series twists.

Findings

Elementary proof of infinitely many irrational odd zeta values

Generalization of hypergeometric series twists for number theory

Simplification of previous complex proof techniques

Abstract

In this small note, we provide an elementary proof of the fact that infinitely many odd zeta values are irrational. For the first time, this celebrated theorem been proven by Rivoal and Ball--Rivoal. The original proof uses highly non-elementary methods like the saddle-point method and Nesterenko's linear independence criterion. Recently, Zudilin has re-proven a slightly weaker form of his important result that at least one of the odd zeta values $\zeta(5),\zeta(7),\zeta(9)$ and $\zeta(11)$ is irrational, by elementary means. His new main ingredient are certain 'twists by half' of hypergeometric series. Generalizing this to 'higher twists' allows us to give a purely elementary proof of the result of Rivoal and Ball--Rivoal.