Get prime number identifying function? $$\prod_{prime}(1-\frac{1}{p^s})\; = \frac{1}{\zeta(s)}$$
$$\Rightarrow\;\prod_{n=2}^{\infty}(1-\frac{1}{n^s})^{v(n)}\; = \frac{1}{\zeta(s)}, \;\;\;\;
v(n) = \left\{\begin{array}{l}1,\;n\;is\;prime\\0,\;otherwise\end{array}\right.$$
$$\Rightarrow\;\sum_{n=2}^{\infty}v(n)\cdot \ln(1-\frac{1}{n^s})\;=-\ln\zeta(s)$$
So by letting s = 2, 4, 6, 8, 10,...
$$\ln(1-\frac1{2^2})\cdot v_2\;+\;\ln(1-\frac1{3^2})\cdot v_3\;+\;\ln(1-\frac1{4^2})\cdot v_4\;+\;...\;=-\ln\frac{\pi^2}{6}$$
$$\ln(1-\frac1{2^4})\cdot v_2\;+\;\ln(1-\frac1{3^4})\cdot v_3\;+\;\ln(1-\frac1{4^4})\cdot v_4\;+\;...\;=-\ln\frac{\pi^4}{90}$$
$$\ln(1-\frac1{2^6})\cdot v_2\;+\;\ln(1-\frac1{3^6})\cdot v_3\;+\;\ln(1-\frac1{4^6})\cdot v_4\;+\;...\;=-\ln\frac{\pi^6}{945}$$
and so on.
Here is my point: this is a set of 1st order equations with unknowns
$v_2, v_3,v_4,v_5,...$ And since they are just first-ordered equations, it is actually possible to solve for $v$!
But I have no idea. Anybody that has ideas? 
Also tell me if you think there are "obstacles" that can't make this happen. Thanks. 
 A: Let $Z_m(s) = -\sum_{i=1}^m \log(1-p_i^{-s})$. You'll recover the primes from $\log \zeta(2k)-Z_m(2k) \sim p_{m+1}^{-2k}$ so that
$$p_{m+1} = \lim_{k \to \infty}(\log \zeta(2k)-Z_m(2k))^{-1/2k}, \qquad Z_{m+1}(s) = Z_m(s)-\log(1- p_{m+1}^{-s})$$

Equivalently let $P(s) = \sum_{p} p^{-s} $ then $$\log \zeta(s)= \sum_{n=1}^\infty \frac{P(ns)}{n} \implies P(s) = \sum_{n=1}^\infty \frac{\mu(n)}{n}\log \zeta(ns)$$
Obtaining that
$$P(2k) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log \zeta(2kn)=\sum_{n=1}^\infty \frac{\mu(n)}{n} \log[(-1)^{kn+1}\frac{B_{2kn}(2\pi)^{2kn}}{2(2kn)!}]$$
And you'll recover the primes from $$P_m(s) = \sum_{i=1}^m p_i^{-s}, \qquad p_{m+1} = \lim_{k \to \infty} (P(2k)-P_m(2k))^{-1/2k}, \qquad P_{m+1}(s) = P_m(s)+ p_{m+1}^{-s}$$
As you see, this is not very different and much simpler to using the Sieve of Eratosthenes.
A: Solving a set of linear equations is easy enough, but here you have an infinite set of equations in infinitely many unknowns. Such a problem can in general not be solved.
