Consider the following sum :

$$S(p,q;s)=\sum_{n=1}^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$

Here , $p$ is a variable w.r.t which we are going to analyse the sum.

$s$ is another parameter with domain $s\in(0,1]$.

I tried to use Abel - Plana summation formula (ABSF) for partial sum :

we can get some crude estimates on integral :

$$I_1(q)= \int_1^q\frac{\sin^2(\frac{p\Gamma(t)}{t})}{t^s}dt $$

as $q\rightarrow \infty $

( Integrand oscillates very wildly in right half plane )

But it seems that second integral in ABSF is impregnable .

Second integral in ABSF:

$$I_2(x)=\int_0^\infty \frac{F(x + iy,s) − F(x − iy,s)}{e^{2πy}-1}dy$$

Where , $F(z)=\frac{\sin^2(\frac{p\Gamma(z)}{z})}{z^s} $

I tried to get estimate on this as $x\rightarrow \infty $ but in vain .

The reason I chose ABSF is that $\Gamma$ is a 'nice' function in terms of complex variables and due to this I'm optimistic about the $I_2$

The importance of this function lies in the fact that , See for ex . for $p=π/2$ the $\sin²$ term is finite for primes and zero for non primes . So I mentioned 'sharp' for this purpose . I need critical details of $I_2$.

( I'm just following the advice of F.R.Villegas to generalize the series with such parametrization . )

If this could be achieved then we are able to get the estimate using purely analytic information ( no number theoretic information like Euler product ) . And this seems (although extremely hard but,) possible as due to relatively elementary nature of integrals .

Question : Can we get an 'Sharp' estimates on the sum w.r.t parameters $p$ and $s$?

(Also , I calculated various values of $I_2(x)$ for various $x$'s and test parameters .)