Consider the following sum:
$$S(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$
The summand is zero for non-primes, and finite and non-decreasing for primes
For more details see the key paper: Connes' paper on Wilson's theorem
Question :
Is there any way we can prove infinitude of primes using above series ?
I tried to attack this problem in my previous questions . I just posted this separately new question (without stating any of my work) to start fresh and expecting new methods and views from users .
Any suggestions and comments are welcome .
Possible unified Applications: We can apply it to other primes of special forms whose Infinitude is unknown. (as Γ is nicely analytic):
$$S_2(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)\sin^2\left(\frac{π\Gamma(n+2)}{2(n+2)}\right)$$
See also for details :Towards a new proof of infinitude of primes ( with possible unified application to other primes of special forms whose Infinitude is unknown):