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Consider the following lacunary sum with parameter $x$:

$$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$

As we can see for $x=\frac{\pi}{2}$ the sum becomes$$\sum_p\cos^2\left(\frac{π}{2p}\right)$$

where $p$ runs through all primes.

What are some non trivial properties of $S(x)$?

Can we at least prove infinitude of primes (at $x=\pi/2$) from this ?

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  • $\begingroup$ Forgive my ignorance, but is $\Gamma$ the Gamma function (en.wikipedia.org/wiki/Gamma_function) here? If so then isn't $\Gamma(n) = (n - 1)!$? $\endgroup$
    – Joseph Harrison
    Commented Aug 10 at 2:55
  • $\begingroup$ @JosephHarrison yes $\endgroup$
    – TPC
    Commented Aug 10 at 2:58
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    $\begingroup$ Is it obvious that you get $\sum_p \cos^2(\pi/2p)$ when you take $x = \pi/2$? Do you have a reference? $\endgroup$
    – Joseph Harrison
    Commented Aug 10 at 3:10
  • $\begingroup$ @JosephHarrison I think for target audience for this sort of things on mathoverflow is pretty elementary! $\endgroup$
    – TPC
    Commented Aug 10 at 8:52

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