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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 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\ $.
