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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):

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  • $\begingroup$ So essentially you'd like to show that $S(p)\rightarrow\infty$? for $p\rightarrow\infty$? $\endgroup$
    – Alex R.
    Commented Mar 31, 2020 at 22:38
  • $\begingroup$ @AlexR. Yes,sir $\endgroup$
    – bambi
    Commented Mar 31, 2020 at 22:39
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    $\begingroup$ Not that this isn't an extremely interesting idea, but I honestly don't see a good way to go forward in general unless one pulls out some extremely difficult mathematics. For example, there really isn't that much difference between $S_2(p)$ and $S_3(p)$ but we would expect the first sum to go to infinity while the second sum converges after the first few terms. However, from an analytic standpoint, I couldn't tell you what the difference is between the series (just that there must be one). $\endgroup$
    – QC_QAOA
    Commented May 14, 2020 at 4:27
  • $\begingroup$ @QC_QAOA seeing the wide range of applications, it's worth to pull that extremely difficult mathematics. $\endgroup$
    – bambi
    Commented May 14, 2020 at 7:06
  • $\begingroup$ Got a downvotes! Please explain what's wrong $\endgroup$
    – bambi
    Commented Oct 24, 2020 at 20:28

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