I bet the product
$$
\prod\_{n=2}^\infty\frac 1 {1-n^{-s}},
$$
which is convergent for ${\rm Re}(s)>1$, has been studied before. Can it be analytically extended across the line ${\rm Re}(s)=1$? If so, is there a functional equation?



What is known about this product?
Yes, the inverse of it, $\prod\_{n=2}^{\infty} \frac{n^s-1}{n^s}$
 has been studied in connection with the so-called $k$-almost prime zeta functions of the second kind. 
It can be evaluated to
$$
\prod\_{n=2}^{\infty} \frac{n^s-1}{n^s}=\prod\_{n=1}^{\infty} \frac{(n+1)^s-1}{(n+1)^s}=
\frac{1}{\prod\_{j=1}^{s-1}\Gamma (2-e^{2\pi ij/s})}.
$$
Then one can use the properties for the $\Gamma$-function.   

A reference is the paper by Richard J Mathar, on HARDY-LITTLEWOOD CONSTANTS EMBEDDED INTO
INFINITE PRODUCTS OVER ALL POSITIVE INTEGERS, and the references therein. The connection with the $k$-almost prime zeta function is given by
$$
\prod\_{n=2}^{\infty}\left(1-\frac{1}{n^s}\right)=\prod\_{k=1}^{\infty}\frac{1}{\zeta\_k(s)}
.$$


Edit: I apologize, but $s$ is assumed to be an integer above. But the question has been discussed already, see 
[Values where infinite products of primes and composites are equal](https://mathoverflow.net/questions/53266/values-where-infinite-products-of-primes-and-composites-are-equal).


