Riemann Zeta Function as a Multiple Integral

Theorem
For $n \in \Z_{> 0}$, the Riemann zeta function is given by:


 * $\ds \map \zeta n = \int_{\closedint 0 1^n} \frac 1 {1 - \prod_{i \mathop = 1}^n x_i} \prod_{i \mathop = 1}^n \rd x_i$

where $\closedint 0 1^n$ denotes the Cartesian $n$th power of the closed real interval $\closedint 0 1$.