Around Wilson's theorem (Q1788082)

The focus of the article is on the sum \[ s(n,x):=\sum_{k=1}^n\;\sin^2\left(\frac{x\Gamma(k)}{k}\right). \] It is shown that the integer part of \(s(n,\frac{\pi}{2})\) coincides with the prime counting function \(\Pi(n)\). Moreover, if \(\frac{a}{b}\) is an irreducible fraction with \(b>1\), then \[ s(n,\frac{a\pi}{b})\sim\left(\frac{1}{2}-\frac{\mu(b)}{2\phi(b)}\right)\Pi(n). \] It is also observed that \(s(n,x)\) has the Gaussian behavior, where \(s(n,x)\sim\frac{n}{2}\) for almost all \(x\in\mathbb{R}\) in the Lebesgue measure; whereas for generic \(x\in\mathbb{R}\) (of the Baire theory), \(\frac{1}{2}\) is a limit point of the sequence \(\frac{s(n,x)}{\Pi(n)}\).