TITLE: An inverse for Euler's zeta function product formula
QUESTION [10 upvotes]: Of course, Euler proved that the Riemann zeta function can be defined as the analytic continuation of a product over all primes.  
$$\zeta(s) = \prod_{p \in \mathbb{P}}\frac1{1-p^{-s}}$$
It is well known (but not something I understand) that the positions of zeros of the zeta function allows one to make inferences about the asymptotic behavior of primes.  Is this a general phenomenon? Does Euler's transform generalize to products over other subsets of the natural numbers $\mathbb{A}$?  
$$\alpha(s) = \prod_{a \in (\mathbb{A} \subset \mathbb{N})}\frac1{1-a^{-s}}$$
Can one then reverse Euler's transform and derive the generating subset $\mathbb{A}$ completely from the new function's zero set?  More generally, how do properties of the derived function's zeros translate to properties of the generating subset?  
And, specifically for the standard Riemann zeta function, if it was shown that exactly one single zero existed off the critical line, what would its position say about the distribution of primes?

REPLY [5 votes]: As you noted, to even talk about the "zero set" of the Riemann zeta function, one needs to have an analytic continuation to the left of the half-plane $\{\Re(s)>1\}$ in which the Euler product converges. For a generic set $A\subset{\mathbb N}$, it's not even clear that the corresponding Euler product has an anlytic continuation at all. In this case, questions concerning the "zero set" of the corresponding function are ill-posed.
You can search the relevant literature with the terms "Beurling primes" or "generalized primes".