The Zeta Quotient $\zeta(3)/ \pi^3$ is Irrational

Abstract

This note proves that the first odd zeta value does not have a closed form formula $\zeta(3)\ne r \pi^3$ for any rational number $r \in \mathbb{Q}$. Furthermore, assuming the irrationality of the second odd zeta value $\zeta(5)$, it is shown that $\zeta(5)\ne r\pi^5$ for any rational number $r \in \mathbb{Q}$.