Weight grading and cross-weight relations for multiple zeta values

Determine whether the Q-algebra of multiple zeta values decomposes as the direct sum of its weight-homogeneous subspaces, equivalently whether there are no linear relations among multiple zeta values of different weights.

Background

For each weight k, the paper defines the Q-vector space of weight-k multiple zeta values and notes that the expected direct-sum decomposition would mean that the weight grading is genuine rather than merely a filtration. This is a longstanding structural question about linear relations among multiple zeta values.

The paper explicitly states that this problem appears to be beyond current methods, so it is not resolved by the results presented for multiple Eisenstein series.

References

The celebrated Goncharov--Zagier direct sum conjecture predicts that there should be no nontrivial $\mathbb{Q}$-linear relations among MZVs of different weights:

— On Multiple Eisenstein Series in Positive Characteristic: Direct Sum Result  (2609.18777 - Chang et al., 16 Sep 2026) in Section 1, Subsection 1.1 (Classical story)

Its weight is $k_1+\cdots+k_r$ and if $\mathcal{Z}k$ denotes the $Q$-span of all multiple zeta values of weight $k$, with $\mathcal{Z}_0=Q$, then it is conjectured that $\mathcal{Z}\stackrel{?}{=}\bigoplus{k\geq0}\mathcal{Z}_k$, where $\mathcal{Z}$ is the $Q$-algebra of multiple zeta values, i.e. conjecturally there are no linear relations among multiple zeta values of different weights. For multiple zeta values this seems to be out of reach with current methods.

— The $\mathfrak{sl}_2$-algebra structure of multiple Eisenstein series  (2609.03777 - Bachmann et al., 3 Sep 2026) in Section 1, Introduction