One of the Odd Zeta Values from $\zeta(5)$ to $\zeta(25)$ Is Irrational. By Elementary Means

TL;DR

This paper presents the first elementary proof that at least one of the odd zeta values between 05 and 25 is irrational, using only basic tools like the prime number theorem and Stirling's approximation.

Contribution

It introduces a new elementary method to prove the irrationality of at least one odd zeta value within a range, avoiding complex techniques like saddle-point methods.

Findings

Proves at least one odd zeta value between 05 and 25 is irrational

Uses only elementary tools such as the prime number theorem and Stirling's approximation

Provides a more accessible proof compared to previous non-elementary methods

Abstract

Available proofs of result of the type 'at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are however counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result - a proof whose technical ingredients are limited to the prime number theorem and Stirling's approximation formula for the factorial.