Consider the following functional :
$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])}{z^s} $.
Let us restrict $s\in[0,1]$
Can we get sharp numerical asymptotic of $I(x)$ as as $x\rightarrow \infty$?
Also, can we get quantitative upper and lower bound estimations on the functional ?
Not trivial : see:
https://math.stackexchange.com/q/3570663/702232
Construction of a certain weight for a functional to satisfy given condition:
