[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform. I myself am very skeptical about this but I want to know, from the experts' point of view, why this approach is not practical (if it is).]
Consider the following partial sum :
$$S(t)=\sum_{n=2}^t\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$
As we can see, the summand is zero for non-primes greater than 5 and finite non-decreasing for primes (see Alain Connes paper on Wilson's theorem.) So,we can use this as prime counting function in some approximate sense.
Due to analyticity of gamma function (and it's decay wrt imaginary part of complex variable) I treated this summation with Finite version Abel-Plana Summation Formula (APSF) (as in Olver's book "Asymptotics and special functions") (summand follows all the properties required by APSF) :
$$\begin{align}f(x) = {} & \sin^2\left(\frac{π\Gamma(x)}{2x}\right)\\ \sum_{k=2}^t f(k)= {} & \frac{f(2) +f(t)}2 + \int_2^t f(x) \, dx \\ & {}+ i\int_0^∞\frac{f(2+iy) − f(2−iy)}{e^{2πy }− 1} \, dy +i \int_0^∞\frac{f(t-iy) − f(t+iy)}{e^{2πy }− 1} \, dy\end{align}$$
Here the first integral $∫f(x)dx$ is okay (highly oscillatory but we can do something: at least numerically)
Now, to analyze the following integral
$$i \int_0^∞\frac{f(t-iy) − f(t+iy)}{e^{2πy }− 1} \,dy ,$$
I tried to get a growth condition on it without much success.
So, I tried to attach a weight such that this integral is convergent to zero as $t\rightarrow\infty$ :
$$F(z) = \omega(z)\sin^2\left(\frac{π\Gamma(z)}{2z}\right)$$
The following three condition should hold for $ω(z)$ :
$$ω(z)>\frac{1}{z}, \quad \forall z∈R$$ and it is analytic for $\Re(z)>2$
(More generally this condition is added to impose the divergence of $\int_2^{+\infty} F(x)dx$. So $ω(z)$ can even be complex valued for real domain as long as the given integral is divergent (to prove infinitude of prime by this method) we can make various other generalizations.)$$\lim_{y→∞}|F(x±iy)|e^{−2πy}=0$$
$$∫_0^∞|F(x+iy)−F(x−iy)|e^{−2πy}dy<+∞$$ for every $x≥1$ and tends to zero as $x→∞$.
This is to eliminate the tricky 2nd integral in the formula.
We can also use some operators on $f(x)$ with preserving prime counting property in order to eliminate 2nd integral like:
Instead of finding the weight ; consider sin^2 term as function of some other function such that:
Construct a generalized function such that:
$$F_∗(z)=\varphi(\sin^2[πΓ(z)/(2z)])$$
$\varphi(x)=0$ 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) )conditions (2) and (3) above holds for such function
Possible candidate: I'm studying this test function for above analysis(at least numerically):
$ F(z,s) = \dfrac{1}{z^s\Gamma(\sin^2[π\Gamma(z)/(2z)])} $.
And let us restrict $s\in[0,1]$
My question :
Is this approach worth further study? If anything wrong with the analysis, please state. Any comments in the directive manner are welcome.
Can we get an estimate on summation $S(t)$ using the above method? ( Estimate as strong as prime number theorem or more stronger)
At its most generality, what can we derive from this approach which is already known about prime numbers (or more than what we know)?
(I've done verious generalizations of this approach but I'm going to keep this short )
Moreover we can generalize this approach as follows :
Consider the following function :
$$T(x)=w(x)\frac{\sin^2(c \frac{\Gamma^2(x)}{x})}{\sin^2(\frac{c}{x})}$$
As we can see, for $c=π$ and finite $w(x)$, $T(x)$ is 0 for non-prime $x$ and 1 for prime $x$.
Now if we consider $w(x)$ as rational function s.t. $\sum_{p}w(p)$ converges and second (complex) integral (Only $T(t)$ instead of $f(t)$) in above analysis converges to zero as t tends to infinity.
Then we can prove the infinitude of primes by computing the first real integral and proving that it is irrational (extremely hard).
I'm very much skeptical about this approach, but I thought about it so i added a description.
The reason I'm interested in infinitude of primes is that; if possible ,such method can be extended to twin primes and other forms of primes whose infinitude is unknown.
Possible Unified Applications: We can apply it to other primes of special forms whose Infinitude is unknown. (as Γ is nicely analytic).
Example:
$$S_2(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)\sin^2\left(\frac{π\Gamma(n+2)}{2(n+2)}\right)$$
A question with same essence is on Math Stack Exchange for more than a year with no response. I also corresponded with various professors about this with not much of help from anyone :
