Mixed Tate motives over $\mathbb{Z}$

Abstract

We prove that the category of mixed Tate motives over $\mathbb{Z}$ is spanned by the motivic fundamental group of $\Pro^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\Q$-linear combination of $\zeta (n_1,...,n_r)$ where $n_i\in \{2,3\}$.