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 $\closedint 0 1$.

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

$=$

$\ds \int_{\closedint 0 1^n} \sum_{j \mathop = 1}^\infty \paren {\prod_{i \mathop = 1}^n x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^\infty \int_{\closedint 0 1^n}\prod_{i \mathop = 1}^n {x_i}^{j - 1} \prod_{i \mathop = 1}^n \rd x_i$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \int_0^1 x^{j - 1}_i \rd x_i$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \intlimits {\frac {x^j} j } 0 1$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^\infty \prod_{i \mathop = 1}^n \frac 1 j$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^n}$

$\ds $

$=$

$\ds \map \zeta n$

$\blacksquare$