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 $\sin^2$ 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 estimates on primes 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 summand and integrals .

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

Can we prove the divergence of series (as a whole) for specified conditions on $p$?

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

Edit:

More generally we can consider the following:

$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$

Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.

The following two conditions should meet for $\omega(z)$:

1. $$\lim_{ y→∞}|F(x ± iy)|e^{−2πy }= 0$$

2. $$\int_0^\infty |F(x + iy) − F(x − iy)|e^{−2πy} dy<+\infty$$ for every $x≥1$ and tends to zero as $x\to\infty$.

How to construct $\omega(z)$?