On Wikipedia there's an OK discussion of multiple zeta values (MZV). We have an identity:

$$ \zeta(2,2) = \sum_{m > n > 0} \frac{1}{m^2 \, n^2} = \frac{3}{4} \sum_{n > 0} \frac{1}{ n^4} = \frac{3}{4} \zeta(4) $$

This formula might already be on the Math.Stackexchange site (link?) Also notice that somehow the zeta function turns partly into an element of $\mathbb{Q}$. I have already found an argument using one of the special cases of the shuffle formula.

$$ \zeta(2)^2 = 2 \, \zeta(2,2) + \zeta(4) $$

Certainly, there still a search for *even more* $\zeta$-functions of this kind. Both sides of this equation have integral formulas (due to Drinfiel'd):

\begin{eqnarray*} \zeta(2,2) &\stackrel{?}{=}& \int_{1 > x_1 > x_2 > x_3 > x_4 > 0} \frac{dx_1}{x_1} \wedge \frac{dx_2}{1-x_2} \wedge \frac{dx_3}{x_3} \wedge \frac{dx_4}{1-x_4} \tag{$*$}\\ \\ \zeta(4) &\stackrel{?}{=}& \int_{1 > x_1 > x_2 > x_3 > x_4 > 0} \frac{dx_1}{x_1} \wedge \;\; \frac{dx_2}{x_2} \;\;\wedge \frac{dx_3}{x_3} \wedge \frac{dx_4}{1-x_4} \tag{$**$} \end{eqnarray*}

Even now, I'm not totally convinced the integrals on the right side, represent the infinite series on the left side. But also, I wonder if there's a stronger notion of equivalence than just that they evaluate to the same number. We have just shown that $\zeta(2,2)$ and $\zeta(4)$ are **periods** These notes of José Ignacio Burgos Gil and Javier Fresán indicate the following:

From the modern point of view, periods appear when comparing de Rham and Betti cohomology of algebraic varieties over number fields.

So, my other question is whether these two 4-forms are the same. Possibly over $\mathbb{C} \backslash \{ 0,1\}$ or $\mathbb{P}^1 \backslash \{ 0,1, \infty\}$ :

\begin{eqnarray*} \omega_1 &=& \frac{dxt_1}{x_1} \wedge \frac{dx_2}{1-x_2} \wedge \frac{dx_3}{x_3} \wedge \frac{dx_4}{1-x_4} \\ \\ \omega_2 &=& \frac{dx_1}{x_1} \wedge \;\; \frac{dx_2}{x_2} \;\;\wedge \frac{dx_3}{x_3} \wedge \frac{dx_4}{1-x_4} \end{eqnarray*}

And I would ask is there a sense in which $\omega_1 = \omega_2$? Mostly I just want help with the integrals.