Asymptotic of highly oscillatory summand $\sin^2\left(\frac{\Gamma(n)}{n}\right)$

Question

How to get an asymptotic of following sum :

$$f(x) = \sum_{n=1}^x \sin^2\left(\frac{\Gamma(n)}{n}\right).$$

Note: Quick investigation tells Classical Euler summation fails.

Answer 1

Let $a_n = \Gamma(n)/n$. Of course $\sin^2(a_n) = 1/2 -\cos(2 a_n)/2$. I would expect that for all positive integers $\ell$, $$\frac{1}{x} \sum_{n=1}^x \exp(i \ell a_n) \to 0 \ \text{as}\ x \to \infty$$ which by Weyl's criterion is equivalent to the sequence $a_n$ being equidistributed mod $2\pi$. But I doubt that this is easy to prove.