While reading Eugene Dickson's book History of the theory of numbers I came across following prime counting formulas:
1.
2.
3.
I want the papers by the authors but a quick google search couldn't give me nothing.
It would be very nice if someone could provide me with the links. (pdf if available)
As regards Andreoli's paper see this link. The paper is written in italian. This is a short summary in english.
If $x\in \mathbb{R}$ and different from $x=0,-1,-2.,\dots$ then the function $$\Phi(x)=\sin^2\left(\pi\frac{\Gamma(x)+1}{x}\right)+\sin^2(\pi x),$$ which is the sum of two squares, is zero if and only if $$\begin{cases} \sin(\pi x)=0\\ \sin\left(\pi\frac{\Gamma(x)+1}{x}\right)=0\\ x\not \in \mathbb{Z}\setminus \mathbb{N}^+ \end{cases}\Leftrightarrow \begin{cases} x\in\mathbb{N}^+\\ \frac{(x-1)!+1}{x}\in\mathbb{N}^+ \end{cases} \Leftrightarrow \text{$x$ is a prime}$$ where we applied the Wilson's Theorem. Since $\Phi$ is a analytic in a neighborhood of the segment $[1,n]$ with $n>1$, by the Argument Principle, for any simple contour $\gamma$ passing through $1$ and $n$ (not a prime) and sufficiently close to the cut $[1,n]$, $$\frac{1}{2\pi i}\oint_{\gamma}\frac{\Phi'(z)}{\Phi(z)}\, dz$$ gives the number of primes less than $n$.