I'm trying to find a closed form expression for the following product: $f(k) = \prod \limits_{m=1 \\m \neq k}^{N} \frac{1}{1-(\frac{k}{m})^{\alpha/2}}$.
This can be simplified or approximated into: $f(a) = \prod \limits_{m=1}^{N} \frac{1}{1-a \; m^{-\alpha/2}}$.
A hint or an expression for either would be great. A limit when N tends to infinity can also be useful. Also, it would be helpful if there is any book with tables of products of sequences.