Evaluating $\zeta(4)$ using iterated integrals

Question

I'd like to evaluate $\zeta(4)$ using iterated integrals. We already know the numerical answer, so it remains to set up the integral and do some of the steps. From the recipe of Ihara-Kaneko-Zagier one has: $$ \zeta(4) = \int_{1 > t_1 > t_2 > t_3 > t_4 > 0} \omega(t_1) \omega(t_2)\omega(t_3) \omega(t_4) = \int_{1 > t_1 > t_2 > t_3 > t_4 > 0} \frac{dt_1}{t_1} \frac{dt_2}{t_2} \frac{dt_3}{t_3} \frac{dt_4}{(1-t_4)}$$ Does this look correct? The domain of integration is a simplex (a 5-cell). What are some of the intermediate steps to evaluating this integral?

Answer 1

In pages $8-9$ of my notes it is proved that by evaluating $$ \int_{0}^{1}\frac{\log(x)\log^2(1-x)}{x}\,dx $$ (which, of course, can be unfolded into a multiple integral) in two different ways we have $$ \zeta(4) = \frac{2}{5}\zeta(2)^2 $$ even without knowing the identity $\zeta(2)=\frac{\pi^2}{6}$, which, on its turn, can be proved by essentially squaring the Taylor series of the $\arctan$ or $\arcsin$ functions. Many equivalent approaches are outlined in this historical thread.