# Prime clasfication by some constructive function

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 statment can illustrate the follwoing 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.

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@martini! Thank you so much for your editing work. – vmrfdu123456 Nov 16 '12 at 10:32
Note that $1/(1-g^2)$ is redundant – Norbert Nov 16 '12 at 10:34
I am confused. What is $f$? It is given by the formula, then the formula is true by definition. And if not, how is it defined? – Harald Hanche-Olsen Nov 16 '12 at 10:40
f is given by the formula. of course, formula is true by definition. – vmrfdu123456 Nov 16 '12 at 10:54
For me it seems that in this form, (3) contradicts (1) and (2). – Berci Nov 16 '12 at 11:02