This morning I've been watching documentary about asterorids, in a scene an astronomer explains the so called image subtraction process or pixel subtraction, a mathematical model used in computerized search for asteroids (I think that is based on making two almost simultaneous photos, and subtract pixels of a object: a very bright object or that reflect a lof of light, that does not interest us, to stay with the light, position, of the small object that is close to the first, that is reflecting low light, which is what interests us). Sorry for my english.
It is well known (see for example [1]) that Prime Number Theorem is logically equivalent to $\varphi(x)\sim x$, or too logically equivalent to $\vartheta(x)\sim 1$, where for $x>0$, $\varphi(x)=\sum_{n\leq x}\Lambda(n)$, Mangoldt function $\Lambda(n)$ equals to $\log p$ if $n=p^{a}$ is a prime power and $0$ in other case, and $\vartheta(x)=\sum_{p\leq x}\log p$, where the sum is extended over all primes $\leq x$. With this in mind, do you known the asymptotic behaviour, or you can prove it, for $\sum_{p,q=p+2\leq x}(\log(p) +\log (p+2))$, where $p$ and $q$ are twin primes? I don't claim a conjecture but I plotted the graphic with these (poor) computations $10, 100, 1000, 10^{4} , 10^{5}, 10^{6} $ as abscises, versus $6.6234, 48.0215, 366.0642, 3.2056\cdot 10^{3}, 2.5033\cdot 10^{4}, 2.0581\cdot 10^{5}$. And alone I don't conclude with this, too I think that could there be some literature about it.
Question. Can you search literature for asymptotic behaviour of $\sum_{p,q=p+2\leq x}(\log(p) +\log (p+2))$, or do a more rigorous computational study. Too I accept detailed hints.
My only goal is edit a nice post in this Mathematics Stack Exchange with your help, if my question is interesting for you. Thanks in advance.
References:
[1] Tom M. Apostol, Introduction to Analytic Number Theory, Springer (1979), p. 79.
The infinitude of the twin primes is an open problem, so currently proving anything about the asymptotics of this function is out of reach.
However, the first Hardy-Littlewood conjecture would imply that your sum is asymptotic to $$4\Pi_2 \frac{x}{\log x}$$ where $$\Pi_2=\prod_{p\geq 3} \frac{p(p-2)}{(p-1)^2}$$ is the twin prime constant.