# A sum involving twin primes and Prime Number Theorem

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.

• If we know a function $f(x)$ such that $\sum_{p,p+2\leq x}\log(p+2)+\log(p)\sim f(x)$ we could solve the twin primes conjecture, because the series diverges iff there are infinitely twin primes and converges iff there is finitely many twin primes. – Marco Cantarini Jul 21 '15 at 9:01
• First sorry by unusual first paragraph, and thanks for your comment. To treat with this question, we have to put the prime twin conjecture (there are infinitely many or there are finitely many twin primes) to work. I bring here the first paragraph because that's how I got the idea, consider this type of sums. – user243301 Jul 21 '15 at 11:37
• @Marco Cantarini Currently I do not maintain contact with university world, and I consider that type of problem could have been already studied , so I ask references. On the other hand also I ask if someone can make a graph for large values. – user243301 Jul 21 '15 at 11:39
• The curiosity of the first paragraph, is that by previous two known arithmetic means and the theorem which in page 79 of Apostol's book, is that by removing the elements that omits the second sum from the first when we measure positions of these constellations with $(1/x)\sum_{n\leq x} \log(x)$ , the light (the truth) that we get is the so called Prime Number Theorem . You do not have to answer to this unusual comment. Thanks Cantarini . – user243301 Jul 21 '15 at 11:39
• A misprint $\vartheta(x)\sim x$. – user243301 Jul 21 '15 at 13:42

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.