Consider the following function:

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

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

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

1. $$\omega(z)>\frac{1}{z},\ \forall z\in\mathbf{R}$$

( More generally this condition is added for divergence of $\int_0^\infty F(x)dx$ So ,  $\omega(z)$  can even be complex valued for real domain as long as the given integral is divergent )

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

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


>>Question : Explicit construction of  $\omega(z)$.

(Is it even possible?)

If such construction of such weight is not possible; then consider another question: 

 One can also ask for the following :


>>$$\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy=o(g_1(x))=O(g_2(x)) $$ 


As $x \to \infty$.



>>what are some possible candidates for $g_1(x)$ and $g_2(x)$ ?Also can we find $g_1(x)$ and $g_2(x)$ for which bounds are sharp?


Here , $f(z) =\sin^2\left(\frac{c\Gamma(z)}{z}\right)$



See [this MSE post][1] and [this MSE post][2] for more details.

[1]:https://math.stackexchange.com/q/3570663/702232

[2]:https://math.stackexchange.com/q/3689175/702232