How to prove or justify the following:
$ f(g)= \frac1{1-g^2} \prod_{k=1}^{\infty} \left(\frac{\sin(\pi \frac gk)}{\pi \frac gk} \cdot \frac 1{1-\frac{g^2}{k^2}}\right). $ The above statement can illustrate the following facts:
(1) if $f(g) = 0$, then $g$ is composite number
(2) If $f(g)$ is not equal to $0$, the $g$ is prime number
(3) if $f(1+g) = 0$, then $g$ is prime.
Please justify.