Proof of Test of primality from E.Dickson's book "History of theory of numbers vol.1" E.Dickson mentioned the following result by E.Zondadari in his book "History of theory of numbers vol.1"(Chap XVIII):
Consider the following 'function':
$$ \frac{\sin²(πx)}{(πx)²(1-x²)^2} \prod_{n=2}^\infty \frac{(πx)} {n\sin(πx/n)}  $$
It's zero if |x|= prime and else otherwise .


How to prove the above identity ?


 A: A copy of Zondadari's paper can be found here.
The key to understanding the stated result is an earlier one that the infinite product
$$
P(x)=\prod_{n=1}^\infty \frac{\sin\frac{\pi x}{n}}{\frac{\pi x}{n}}
$$
converges absolutely and uniformly to an analytic function whose zeros are precisely the non-zero integers, with multiplicities equal to the number of their positive divisors. By $\ \frac{\sin\frac{\pi x}{n}}{\frac{\pi x}{n}}\ $ Zondadari obviously here means the analytic function 
$\ \sum_\limits{i=0}^\infty \frac{(-1)^i}{(2i+1)!}\left(\frac{\pi x}{n}\right)^{2i}\ $, which assumes the value $1$ at $\ x=0\ $, and adopts the convention that an infinite product with only a finite number of zeroes and a tail that converges to a non-zero quantity is convergent.
Enumerating the zeros of this product, we have $\ \pm 1\ $, each of multiplicity $1$, primes (positive or negative), each of multiplicity $2$, and and composites, each of multiplicity $\ d\ $, where $\ d\  $ is the number of its positive divisors (equal to $\ \left(n_1+1\right) \left(n_2+1\right)\dots \left(n_r+1\right)\ $, where $\ p_1^{n_1}p_2^{n_2}\dots p_r^{n_r}\ $ is its prime decomposition).
Now the function
$$
f(x) =\frac{\sin^3\pi x}{(\pi x)^3\left(1-x^2\right)^2}
$$
has zeros of multiplicty $3$ at every integer, except $0$, where it has value $1$, and $\ \pm1\ $, where it has zeros of multiplicity $1$.  Therefore, the function
\begin{align}
g(x)&= \frac{\sin^3\pi x}{(\pi x)^3\left(1-x^2\right)^2}\cdot\frac{1}{P(x)}\\
&= \frac{\sin^2\pi x}{(\pi x)^2\left(1-x^2\right)^2} \prod_{n=2}^\infty \frac{\pi x}{n\sin\frac{\pi x}{n}}
\end{align}
has zeros of multiplictity $1$ for prime $\ x\ $, a pole of multiplicity $\ d-3\ $ at any composite $\ x\ $ with $\ d\ $ positive divisors, and has a finite non-zero value at $\ x=0\ $ and $\ x=\pm1\ $.
