Zagier’s dimension conjecture for multiple zeta values

Determine whether the dimension of the Q-vector space spanned by multiple zeta values of weight N satisfies \(\dim_{\Q} \mathcal{Z}^1_N=d_N\), where \(d_N\) is defined by \(d_N=d_{N-2}+d_{N-3}\) with \(d_0=1\), \(d_1=0\), and \(d_2=1\).

Background

The paper defines $\mathcal{Z}^1_N$ as the $\Q$-vector space spanned by multiple zeta values of weight $N$. Zagier conjectured an exact formula for its dimension in terms of the recursively defined sequence $d_N$.

The text notes that Terasoma and Deligne–Goncharov established the upper bound $\dim_{\Q}\mathcal{Z}^1_N\le d_N$, but the stated equality remains the unresolved part of the conjecture. This problem concerns determining the precise algebraic dimension of the space of ordinary multiple zeta values in every weight.