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

Question

I'm trying to analyse the primes with the following point of view

Consider the following partial sum :

$$S(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$

The summand is zero for non-primes greater than 5 , and finite and non-decreasing for primes (see Connes paper on Wilson's theorem)

I treated this sum with Finite version Abel-Plana Summation Formula (APSF) (as in Olver's book "Asymptotics and special functions")

\begin{align}f(x) = {} & \sin^2\left(\frac{π\Gamma(x)}{2x}\right)\\ \sum_{k=2}^p f(k)= {} & \frac{f(2) +f(p)}2 + \int_2^p f(x) \, dx \\ & {}+ i\int_0^∞\frac{f(2+iy) − f(2−iy)}{e^{2πy }− 1} \, dy +i \int_0^∞\frac{f(p-iy) − f(p+iy)}{e^{2πy }− 1} \, dy \end{align}

Here the first integral $\int_2^p f(x) \,dx$ is okay (highly oscillatory but we can do something: at least numerically) ( numerical analysts are welcome to provide graphs of this for large ( at least $10^4$) $p$. I am unable to do so in Mathematica.)

See on computational science SE

$$i \int_0^∞\frac{f(p-iy) − f(p+iy)}{e^{2πy }− 1} \,dy$$

This integral is very tricky, I tried to get growth condition , upper and lower bounds on it but in vain.

Then, I tried to attach a weight such that:

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

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

The following three condition should hold 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 $x\to\infty$.

This is to eliminate the tricky 2nd integral in the formula.

I can't find a weight that satisfies this; nor do I know if it is even possible to find one (!?)

(1) Is there any other way to eliminate this second integral as $p\rightarrow\infty$?

(2) Is there a better summation technique to analyse this (type of) problem?

(3) Can we twist (the frequency part of)/change $F(x)$ to make the second integral more sane? (i.e. to make the magnitude of $f$'s imaginary part satisfy condition 3)

Note : I know these type of trig primality-tests are not practical but this one interests me so .....( interest is due to the fact that $\Gamma$ is nicely analytic).

Also this could provide a new insights to deal with primes ( if it's workable at all).

If argument can't work please explain why(?).

UPDATE :

Instead of finding the weight ; I considered $\sin^2$ term as function of some other function such that:

Construct a generalized function such that:

$$ F_*(z, s) = \dfrac{\phi(\sin^2[π\Gamma(z)/(2z)])}{z^s} $$

(1) $\phi(x) =0$ 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

A very 'close' example :

$$ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])}{z^s} $$

Let us restrict $s\in[0,1]$

Hence ,

Now ,

$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$

Questions remain:

Can we get 'sharp' numerical asymptotic of $I(x)$ as as $x\rightarrow \infty$?

Also, can we get quantitative upper and lower bound estimations on the functional ?

Also, other possible candidates for $\phi$? And henceforth the above analogous analysis as above ?

See: https://mathoverflow.net/q/354962/145581

Related Question : Infinitude of primes using series introduced in Connes' paper on Wilson's theorem

Possible Unified Applications: We can apply it to other primes of special forms whose Infinitude is unknown. (as Γ is nicely analytic).

$$S_2(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)\sin^2\left(\frac{π\Gamma(n+2)}{2(n+2)}\right)$$

For more details see : On a growth condition satisfied by given functional : Also in this post there is list of values of functional for small X's .

Any Comments from numerical methods- experts are welcome (numerical analysis of first and second integral for large (at least $10^2$) values of respective variable ):

See related numerical-method based question : Comparison of integrals with a function (at least numerically):

One sentence summary question: How to get hold of second integral ( growth conditions, roots and other properties ) so that we can use it for our purpose?

Or

How to get rid of second integral using a weight or composite function method described above ?

I modified second integral to various forms to make it workable with given conditions . But if you have a version that works with given conditions as mentioned please add and explain .

As one can see I have various doubt about this approach. But I need some expert comments with technical details why this approach is less likely workable.