[Francis Brown Theorem](http://www.ihes.fr/~brown/Arbeitstatung.pdf) says that $\zeta(a\_{1},\dots ,a\_{r})$ the multi-zeta value of weight $N=a\_{1}+\dots +a\_{r}$ is a $\mathbb{Q}$-linear combination of elements of the set $S=\{\zeta(a\_{1},\dots, a\_{n})| \text{for any i, } a\_{i}\in\{2,3\} \text{ and } a\_{1}+\dots +a\_{n}=N \}$


What I don't understand in the statement is the following: are the elements of $S$ $\mathbb{Q}$-linearly independent? 


I read that there is a surprising $\mathbb{Q}$-linear relation between some values of the multi-zeta function $\zeta$ of weight 12. The relation I'm talking about is the following:
$$28\zeta(3,9)+ 150\zeta(5,7)+168\zeta(7,5)= \frac{5197}{691}\zeta(12) $$ 
Does this relation have some arithmetic-geometric interpretation ?


[In his talk](https://www.youtube.com/watch?v=PbLurA42hxs) (around 5:50) F. Brown made a remark by saying that the relation above relation is related to cusp forms for $SL(2,\mathbb{Z})$. What that means exactly and what is the relation between cusps forms and the $\mathbb{Q}$-linear relation above ? 



Multizeta function values
The elements of $S$ are conjectured to be $\mathbb{Q}$-linearly independent, and so a basis for the $\mathbb{Q}$-linear span of the multiple zeta values. 


This is what Francis Brown accomplished at the motivic level. Actually, his proof via motives goes through the direction that is currently missing on the level of numbers, which is why it doesn't lead to an algorithm for writing down the (conjecturally unique) expression. In Theorem 7.4 of [*Mixed Tate motives over $\mathbb{Z}$* (Ann. Math., 2012)], he proved that the motivic enrichments $\zeta^{\mathbb{m}}(\mathbf{s})$, $\mathbf{s} \in \{2,3\}^{\times}$ of the $\{2,3\}^{\times}$-zeta values are $\mathbb{Q}$-linearly independent. The dimension $d\_n$ of the $\mathbb{Q}$-linear span $\mathcal{M}\_n$ of the motivic MZV of a given weight $n$ is well known to be equal to the $z^n$ coefficient of $1/(1-z^2-z^3)$, which is also the number of expressions of $n$ as a sum of $\{2,3\}$-elements, and so the number of $\{2,3\}^{\times}$ zeta values of weight $n$. The linear independence of the $\zeta^{\mathbb{m}}(\mathbf{s}), \, \mathbf{s} \in \{2,3\}^{\times}$ then implies the equality: these elements form a basis for $\bigoplus\_n \mathcal{M}\_n$. There is a realization homomorphism $\mathcal{M} \to \mathbb{R}$ sending $\zeta^{\mathbb{m}}(\mathbf{s}) \to \zeta(\mathbf{s})$, and it follows (non-constructively) that there is a representation on the level of numbers. 


However, proving $\mathbb{Q}$-linear independence remains completely out of reach on the level of numbers. We know very little here: even the irrationality of $\zeta(5)$ has remained an unsolved problem (for instance, see [this MO question and answer](https://mathoverflow.net/questions/257988/values-of-zeta-function-are-linearly-independent/257991#257991)).


What does follow immediately from the motivic formalism is the upper bound $\dim{M\_n} \leq d\_n$ of the conjectured dimension of the $\mathbb{Q}$-linear span of the weight $n$ MZV. This was proved independently by Terasoma and Goncharov; it means that there are many linear relations among MZV of a given (high enough) weight. Given that, identities like the weight $12$ one you quote should not come as a surprise. Explicit such identities were obtained by [Gangl, Kaneko and Zagier](http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/GKZ.pdf). They are linked to modular forms of weight $n$, via Eisenstein series. 


