Zagier's dimension conjecture for multiple zeta values

Let $\zeta(s_1,\ldots,s_r)$ be a multiple zeta value with $s_1>1$ and $s_2,\ldots,s_r\ge1$, and let its weight be $s_1+\cdots+s_r$. For $k\ge1$, define

$$\mathcal Z_k:=\operatorname{span}_{\mathbb Q}\{\zeta(s_1,\ldots,s_r):r\ge1,\ s_1>1,\ s_2,\ldots,s_r\ge1,\ s_1+\cdots+s_r=k\}$$

and $d_k:=\dim\mathcal Z_k$. Zagier's dimension conjecture. One has $d_1=0$, $d_2=d_3=1$, and

$$d_k=d_{k-3}+d_{k-2},\qquad k\ge4.$$

This predicts the dimensions of the weight-graded pieces of the space of multiple zeta values. In particular, the conjectural numbers $d_k$ count the independent generators in each weight, and there are no linear relations between elements of different weights; the source presents this as a conjectural description supported by numerical checks.