# Asymptotic of given functional as $x\rightarrow\infty$

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 ?

Moreover , if someone can construct a generalized function such that: $$F_*(z, s) = \dfrac{\phi(\sin^2[π\Gamma(z)/(2z)])}{z^s}$$

(1) $$\phi(x) =0$$ if $$x$$ is zero ; and finite otherwise

(2)$$I_*(x,s)\rightarrow c$$ as $$x\rightarrow\infty$$

Here $$c$$ is a constant .

• Any updates on the question? – Bambi Mar 17 at 11:18