$\zeta(5)$ is irrational

TL;DR

This paper provides an elementary proof demonstrating that the value of the Riemann zeta function at 5, denoted as ζ(5), is irrational, using basic number theory tools.

Contribution

It offers a novel elementary proof of ζ(5)'s irrationality, simplifying previous complex approaches.

Findings

Confirmed ζ(5) is irrational.

Utilized Dirichlet's approximation theorem and Prime Number Theorem.

Simplified proof accessible to a broader mathematical audience.

Abstract

We present an elementary proof of the irrationality of $\zeta(5)$ based upon the Dirichlet's approximation theorem and the Prime Number Theorem.