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  $\omega(z)$.

If such construction of such weight is not possible; then consider the following question 

 Prove if following growth exists  :


>>$$\int_0^\infty \frac{F_1(x + iy) − F_1(x − iy)}{e^{2πy}-1} dy=o\left(\int_2^x F_1(t) dt\right) $$ ?

Here , $F_1(z) =\sin^2\left(\frac{c\Gamma(z)}{z}\right)$

( I have a rough, non-rigorous argument which asserts the given growth but I want rigorous argument for confirmation )

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

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