Consider the following lacunary sum with parameter $x$:
$$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right)$$$$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$
As we can see for $x=\frac{π}{2}$$x=\frac{\pi}{2}$ the sum becomes$$\sum_p\cos^2\left(\frac{π}{2p}\right)$$
Where pwhere $p$ runs through all primes.
What are some non trivial properties of $S(x)$?
Can we atleastat least prove infinitude of primes (at $x=π/2$$x=\pi/2$)from from this ?
