Let $\pi_2(x)$ the twin prime-counting function that counts the number of twin primes $p,p+2$ with $p\leq x$, and $C_2$ is the the twin prime constant. We assume the Twin Prime conjecture, see it as Conjecture 3, the definition of $C_2$ and the figure 1 here in this Sebah and Gourdon, Introduction to twin primes and Brun’s constant computation, (2002) from numbers.computation.free.fr.
I am wondering if is it possible state a similar asymptotic, for twin primes on assumption of the Twin Prime conjecture as I've said in previous paragraph, studied by Rubinstein and Sarnak. The identity that I am saying appears for example in page 221 of Granville and Martin, Carreras de números primos, La Gaceta de la Real Sociedad Matemática Española, Vol 8.1 (2005) (in spanish, there is a version in english edited with title Prime Number Races from the American Mathematical Monthly, at 2006).
I believe that this question is not in the literature but maybe is related to the calculation in the Apéndice II of previous paper from La Gaceta.
Question. We assume previous Twin Prime conjecture and we imagine that our purpose should be write a similar statement for twin primes than Sarnak and Rubinstein wrote for prime numbers. Then I believe that the first step should be:
Calculate the asymptotic behaviour of $$\sum_{x\leq X:\pi_2(x)<2C_2\int_2^x\frac{dt}{\log^2t}}\frac{1}{x},$$ as $X\to\infty$. Can you provide the calculations or hints to get this asymptotic behaviour?
Reference:
[1] Hardy and Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. Vol. 44 (1923).
Your question is a natural one, and I'm not sure why this has been downvoted.
While it may be true the the logarithmic density exists, we have little to no idea how to prove this, even after assuming many conjectures about the zeros of the Riemann Zeta function. To understand why, let's look at the ingredients in Rubenstein and Sarnak's theorem. They need to assume the validity of two major conjectures:
The Riemann Hypothesis is equivalent to having a very strong bound on the error term when counting primes, that is RH is equivalent to $$|E(x)|\ll_\epsilon x^{\frac{1}{2}+\epsilon}$$ for any $\epsilon>0$, where $$E(x)=\pi(x)-\text{li}(x).$$ The Linear Independence Hypothesis assumes that all of the imaginary parts zeros of zeta are independent over the rationals, and this is used to obtain finer grain distributional information about the behavior of $E(x)$. Putting these together, they can obtain results for the logarithmic density of the set of points where $\pi(x)>\text{li}(x)$.
Twin Primes: Moving on to the problem for twin primes, analogous to $(1)$ we would need to obtain a strong bound on the size of the error term $$|E_2(x)|=\left|\pi_2(x)-2C_2\int_2^x \frac{1}{\log^2 t} dt\right|,$$ and analogous to $(2)$ we would need a very strong assumption that allows us to obtain distributional information for the size of the error term. However, unlike in the case for primes, we do not even know how to provide $(1)$ - we do not know what the size of this error term should be, even under the assumption of the Riemann Hypothesis or the Generalized Riemann Hypothesis. In fact, we can't even disprove obviously wrong statements such as $$\left|\pi_2(x)-2C_2\int_2^x \frac{1}{\log^2 t} dt\right|<C$$ for some fixed constant $C$. The error term should almost certainly grow like $x^{\frac{1}{2}+o(1)}$ but we don't know how to prove anything here. Given that we are unable to obtain the right size of the error term, distributional information is out of the question.
So while your question is a natural one, currently it is not known how to prove anything close to that, even under the strongest assumptions about the distributions of the zeros of zeta.