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

## TL;DR

This paper proves that the value of the Riemann zeta function at 3, ζ(3), cannot be expressed as a rational multiple of π^3, and under the assumption of ζ(5)'s irrationality, it cannot be expressed as a rational multiple of π^5.

## Contribution

It establishes the irrationality of ζ(3)/π^3 and, assuming ζ(5) is irrational, the irrationality of ζ(5)/π^5, advancing understanding of zeta values' algebraic independence.

## Key findings

- ζ(3) ≠ rπ^3 for any rational r
- Assuming ζ(5) is irrational, ζ(5) ≠ rπ^5 for any rational r
- Provides new irrationality results for odd zeta values

## 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 $r \in \mathbb{Q}$.