*EDIT* There appears to be a numerical zeta function $\zeta(2)$ as well as at least **two** different "motivic" zeta function realizations (Betti and de Rham) $\zeta^{\mathfrak{m}}(2)$. The "period map" of equally mysterious properties relates the two objects and proves the Hoffman conjecture.

Implicit in the comment is that $\zeta(2)$ is a period, which got lifted to a motive... Possibly $\zeta(2)$ has become a homological object? The comments did not explain anything.

I tried to read a paper of Francis Brown Mixed Tate Motives over $\mathbb{Z}$ and I was confused about the status of the conjecture he states:

ConjectureEvery multiple zeta value $\zeta(n_1, \dots, n_r)$ is a $\mathbb{Q}$-linear combination of multiple zeta values at $n \in \{ 2, 3\}$: $$ \big\{ \zeta(n_1, \dots, n_r): \text{ where } n_i \in \{ 2, 3\} \big\} $$

He proves a verion of this using motivic zeta functions, which do not seem to be numerical at all. Brown proves:

TheoremThe set of elements $ \{ \zeta^{\mathfrak{m}}(n_1, \dots, n_r): \text{ where } n_i \in \{ 2, 3\} \} $ are a basis in the space of motivic multiple zeta values.

In fact, I haven't any idea what a motive is. Despite it's elementary appearance, it seems to be related to work of Goncharov and Deligne [1].

Wikipedia's example for the affine line and projectiv line could potentially make sense:

$$ Z(\mathbb{A}^n, t) = \frac{1}{1 - \mathbb{L}^n t} \hspace{0.25in}\text{ and }\hspace{0.25in} Z(\mathbb{P}^n, t) = \prod_{i=0}^n \frac{1}{1 - \mathbb{L}^i t}$$

Wikipedia offers another exampls with the Hilbert scheme of points. None of that seems relevant.

Confusingly Brown, notes that in his conjecture $\zeta^{\mathfrak{m}}(2) \neq 0$ unlike in Goncharov's work.

How does $\zeta^{\mathfrak{m}}(2)$ and relate to $\zeta(2)$ ? Does Brown's Theorem prove the Conjecture (of Hoffman)?

Right now Brown's discussion does not mean very much to me because it rests on some rather difficult concepts:

- Tannakian Categories (and it's Galois Group)
- Mixed Motives
- Motivic Iterated Integrals
- Betti and de Rham realizations of motives
- Why are Lyndon words relevant?
- Do the motivic relations he finds descend to relations of plain zeta values?
- Which "period map" is being used?

My impression is that the Shuffle Relations (which one might find in Kaneko-Ihara-Zagier) get lifted to something very complicated (which could be the Tannakian Category). He has two notes on Feynman Amplitudes [1, 2] which are broader than multiple zeta functions.

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