The linear independence of $1$, $\zeta (3)$, and $\zeta (4)$

Abstract

In a recent breakthrough, Fauzan proved the irrationality of $\zeta (5)$. In this paper, we combine Fauzan's determinant method with classical simultaneous approximations of zeta values studied by Zudilin and Dauguet--Zudilin to prove the linear independence of $1$, $\zeta (3)$, and $\zeta (4)$ over $\mathbb{Q}$. As a corollary, $\zeta (3)/{\pi }^4$ is irrational.

26 pages, 1 table. v2 added some references