Consider the following sum:
$$S(t)=\sum_{n=5}^t\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$
As we can see this approximates $π(t)$ i.e. prime counting function pretty well. For details visit this paper by Alain Connes: https://arxiv.org/abs/1809.02832
This is based on Wilson's theorem.
We can similarly construct " twin prime counting function" as follows:
$$S_2(t)=\sum_{n=5}^t\sin^2\left(\frac{π\Gamma(n)}{2n}\right)\sin^2\left(\frac{π\Gamma(n+2)}{2(n+2)}\right)$$
But now again by Wilson's theorem (twin prime Wilson theorem):
$m,m+2$ are primes iff
$$4(\Gamma(m)+1)=-m(\mod(m(m+2)))$$
So now my question is :
Can we construct an analogues twin prime counting function like prime counting function above using the above twin prime Wilson theorem( i.e. with only one $\sin²(.)$ term?)?
Given Wilson's theorem as stated above: $4(Γ(m)+1)=−m(mod(m(m+2)))$ thus $4(Γ(m)+1)+m=0(mod(m(m+2)))$, thus $4(Γ(m)+1)_mm=nm(m+2)$.
So $\frac{4(Γ(m)+1)+m}{m(m+2)}\in \mathbb{N}$ iff $m$ and $m+2$ are prime.
The function $F(t)=\displaystyle\sum_{m=0}^{t}{\lfloor \sin^2\left( \pi \frac{4(Γ(m)+1)+m}{m(m+2)}\right)}\rfloor $ will then count the number of primes $p$ below $t$ for which the number $p+2$ is also prime.