Consider the following sum :

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

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 :


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

as $q\rightarrow \infty $ 

( Integrand oscillates very wildly in right half plane )

But it seems that second integral in ABSF is impregnable .


>> Question :  Can we get divergence criteria for the sum w.r.t parameters $p$ and $s$?


Note : 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{\Gamma(z)}{z})}{z^s} $

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

(Also , I calculated various values of I_2(x) for various test parameters .)

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 $I_2$