$\Gamma(z)/(2z)$=even integer

Also if condition (3) is not possible in any way, can we get estimate on analogues I_2(x) with use of suitable weight which makes things easier?

$\Gamma(z)/(z)$=even integer

Also if condition (3) is not possible in any way, can we get estimate on analogues $I_2(x)$ with use of suitable weight which makes things easier?

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

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

Can we Explicitly construct $\omega(z)$?

$F(z) = {\phi(\sin^2[π\Gamma(z)/(2z)])}$

(1) ϕ(x)=0 if x is zero ; and 'suitably' finite otherwise (Here , 'suitably' means a value which guarantees the expected divergence of sum (very close to 1 or greater than or equal to 1) )

(2) condition (3) holds for such function.

Could we make above analysis workable?

Also, I think the condition (3) is very hard to achieve because of the complex roots of equations $\Gamma(z)/(2z)$=even integers.

If this could be achieved then we are able to get the estimates on primes using purely analytic information ( no number theoretic information like Euler product ) . And this seems (although extremely hard but,) possible as due to relatively elementary nature of summand and integrals .

Note: I know this question received negative reviews (due to both my behavior and insufficient information ) but please consider the importance of question.

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

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

Can we Explicitly construct $\omega(z)$?

$$F(z) = {\phi(\sin^2[π\Gamma(z)/(2z)])}$$

(1)$ ϕ(x)=0$ if x is zero ; and 'suitably' finite otherwise (Here , 'suitably' means a value which guarantees the expected divergence of sum (very close to 1 or greater than or equal to 1) )

(2) condition (3) holds for such function.

Could we make above analysis workable?

Also, I think the condition (3) is very hard to achieve because of the complex roots of equations

$\Gamma(z)/(2z)$=even integer

Also if condition (3) is not possible in any way, can we get estimate on analogues I_2(x) with use of suitable weight which makes things easier?

If this could be achieved then we are able to get the estimates on primes using purely analytic information ( no number theoretic information like Euler product ) . And this seems (although extremely hard but,) possible as due to relatively elementary nature of summand and integrals .

Note: I know this question received negative reviews (due to both my behavior and insufficient information ) but please consider the importance of question.

Also see: https://math.stackexchange.com/q/3570663/789323

If this could be achieved then we are able to get the estimates on primes using purely analytic information ( no number theoretic information like Euler product ) . And this seems (although extremely hard but,) possible as due to relatively elementary nature of summand and integrals .

Question : Can we get an 'Sharp' estimates on the sum w.r.t parameters $p$ and $s$?

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

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

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

How to construct $\omega(z)$?

Question : Can we get an 'Sharp' estimates on the sum w.r.t parameters $p$ and $s$?

I try to construct the $\omega(z)$ s.t.

$$\sum_2^n F(n)= \int_2^n F(x)dx + A$$

Here A is constant.

For this to be true the following three conditions should meet for $\omega(z)$ in context of APSF:

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

( More generally this condition is added for divergence of $\int_c^\infty F(x)dx$ 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 $x\to\infty$.

Can we Explicitly construct $\omega(z)$?

Even if one can omit 1st condition and able to construct the weight s.t. it follows condition 2,3 please mention. ( i.e in this case the integral $\int_2^\infty F(x)dx$ is convergent.

Also one can generalize further:

$F(z) = {\phi(\sin^2[π\Gamma(z)/(2z)])}$

S.t.

(1) ϕ(x)=0 if x is zero ; and 'suitably' finite otherwise (Here , 'suitably' means a value which guarantees the expected divergence of sum (very close to 1 or greater than or equal to 1) )

(2) condition (3) holds for such function.

Could we make above analysis workable?

Also, I think the condition (3) is very hard to achieve because of the complex roots of equations $\Gamma(z)/(2z)$=even integers.

If this could be achieved then we are able to get the estimates on primes using purely analytic information ( no number theoretic information like Euler product ) . And this seems (although extremely hard but,) possible as due to relatively elementary nature of summand and integrals .

Note: I know this question received negative reviews (due to both my behavior and insufficient information ) but please consider the importance of question.