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.

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

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