A relatively recent paper Alain Connes - Around Wilson's theorem introduced the function $$ S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr). $$ In the same paper, he proved that the integer part of $S(n, \pi/2)$ represents the prime counting function $\pi(n)$.

Can we bound the sum $S(n,x)$, or at least get prime number theorem from $S(n, \pi/2)$?

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    $\begingroup$ It is not news that you can use trig functions and Wilson's Theorem to count primes, see, e.g., primes.utm.edu/notes/faq/p_n.html If such formulas were of any use in deriving the Prime Number Theorem I suspect we would have known about it long ago. $\endgroup$ – Gerry Myerson Aug 29 at 1:10
  • $\begingroup$ @Gerry Myerson I also know that there's various formulas . But they only use zeroes and ones for counting i.e. distributional sense . But I think the above formula is something different than all others; So I've cared to ask $\endgroup$ – Alexander supertramp Aug 29 at 4:44
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    $\begingroup$ I second that said the previous professor (in fact I was tempted to write this comment yesterday). On the other hand see the last paragraph of page 432, or the penultimate paragraph of page 434 from Leonard Eugene Dickson, History of the Theory of Numbers, Volume I Divisibility and Primality, Dover Publications 2005. $\endgroup$ – user142929 Aug 29 at 5:52

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