A note on the Irrationality of $\zeta(5)$ and Higher Odd Zeta Values

Abstract

In this note, we prove the irrationality of $\zeta(5)$ and generalize the method to prove the irrationality of all higher odd zeta values. Our proof relies on the method of contradiction, existence of solution of a system of Linear Diophantine equation, and mathematical induction. For $n\geq 1$, we denote $d_n=\text{lcm}(1,2,...,n)$. In the first part of the article, we assume $\zeta(5)$ is rational, say $a/b$. We observe that for $n\geq b$, there exists a system of equations involving linear combination of $\zeta(5)$ that has a solution. Later using the existence of solution of the Linear Diophantine equation, we show that such a system of linear combination of $\zeta(5)$ has no solution, which is a contradiction. In the second part of the article, we generalise this method for all higher odd zeta values.