Fractional Integration and the Riemann Zeta function

1. Oct 30, 2005

benorin

So it is well-known that for $$n=2,3,...$$ the following equation holds
$$\zeta(n)=\int_{x_{n}=0}^{1}\int_{x_{n-1}=0}^{1}\cdot\cdot\cdot\int_{x_{1}=0}^{1}\left(\frac{1}{1-\prod_{k=1}^{n}x_{k}}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}dx_{n}$$
My question is how can this relation be extended to $$n\in\mathbb{C}\setminus \{1\}$$, or some appreciable subset thereof (e.g. $$\Re(n)>1$$) using fractional integration? Any help, suggestions, or even wanton derision, well, not that :yuck:, but anything useful would be counted kind.
Thanks,
-Ben