Construction/good guess of a certain weight for a functional to satisfy given condition using numerics:

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 )

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

Question : candidates for $$\omega(z)$$.

The question is related to my question: Towards a new proof of infinitude of primes ( with possible unified application to other primes of special forms whose Infinitude is unknown):

The reason for the separate question is that the above post has many sub-questions so this question has only one question.

• I'm not sure why you are asking this on the Mathematica site. – mikado Mar 5 at 18:26