Fractional Calculus and the Riemann Zeta function

[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?

My bad: meant to post this in the Calculus & Analysis forum.

 

[CarlB]

Maybe you could do it by the "dimensional regularization" method that the physicists use to eliminate cancelling infinities in QFT. The idea is that you write the RHS using a method that works for arbitrary dimensions.

For the problem at hand, you might begin by converting the limits of integration to go over the whole real line instead of from 0 to 1. Then convert to spherical coordinates and hopefully write it in a way that eliminates N from the number of integrals, for example, by integrating the angular part.

Do tell us how when you find out.

Carl