At least one of the three numbers $\zeta (5)$, $\zeta (7)$, $\zeta (9)$ is irrational

Abstract

In this article, we prove the irrationality of at least one of the three numbers $\zeta (5),\zeta (7)$ and $\zeta (9)$ is irrational. We find linear forms in $1,\zeta (5),\zeta (7)$ and $\zeta (9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $\Phi$ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $\zeta (5),\zeta (7)$ and $\zeta (9)$ is irrational.