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 $\mathbb{Q}$.