Let $p$ be a prime such that $p+2$ is Also a prime.
Define
$$ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $$
For $z \neq 1 $ and $re(z) > 1$ this $f(z)$ always converges.
If there are infinitely many prime twins , and prime twins grow like $O (n * ln(n)^2 ) $ as is commonly believed , then clearly $f(1)$ diverges.
If there are only finitely many prime twins then $f(z)$ converges everywhere ; it would be an entire function.
For the remainder of this , let us assume there are infinitely many prime twins and they grow like $ O( n * \ln(n)^2 ) $. Thus we assume $f(1)$ diverges.
Notice the similarity with the Riemann zeta function or prime zeta function.
So the main questions are :
- where are the zero’s of $f(z) $ ?
- How about analytic continuations ? Can it be extended to the whole complex plane ? Do we need $re(z) > 0 $ ?
- Do we have a critical line ? Or can a slight modification ( such as neglegting a few twins or a slightly different infinite sum ) lead to a critical line ?
- Is $f(1)$ a pole or singularity ?
- Do the zero’s have real part of 1/2 ? Do any have real part 1/2 ?
Large Numerical data suggests that
$$ \pi_2(n) = \int_2^n c_2 \cdot ln(x)^{-2} dx + O(n^{2+\epsilon}) (**) $$
Where O is big O notation, $c_2 $ is the prime twin constant and $\pi_2$ is the twin prime counting function. This is more or less an analogue of the Riemann hypothesis , which led to question 5) .
Maybe it matters If we assume $(**)$ or not to answer 1-5. The situation is unclear.
Im fascinated by this function. At the same time however, i wonder If this is the best analogue of the zeta function to study prime twins or not. Or the best function to study the twins.
I assume there are no special values known. ( this is not a main question , Just a thought )
Related to questions 1-5 and $(**)$
Is $f(z)(z-1)$ entire ??
——
Update
Tommy1729 sent me a message where he says that according to his estimates
$$ f(z) + 3/2 = 0 $$
Has infinitely many solutions with $ re(z) $ close to $ 1$ and close to $2/3$.
I was able to confirm the observation for twins up to 43 and the 4 first zero’s.
Could those values $1,2/3$ be the exact real parts ?
Do we have 2 critical lines for $ f(z) + 3/2 $ ? And What does that all imply ? What is the meaning of that all ?
——