Is there any known relationship between Goldbach's comet G(n) and the prime counting function (${\pi(n)}$)? The "extended" Goldbach conjecture defines R(n) as the number of representations of an even number n as the sum of two primes, but the approach is not related directly with ${\pi(n)}$, is there any kind of Goldbach-${\pi(n)}$ (I will call it G${\pi(n)}$ for short) function? 

http://mathworld.wolfram.com/GoldbachConjecture.html

I have tried an approach to that idea as follows.


Calculate ${\pi(n)}$
Calculate ${\pi(n/2)}$
I defined then G${\pi(n/2)}$ as the subset of primes p from ${\pi(n/2)}$ that are symmetrical on n/2, so they have a counterpart prime pc = n-p in [n/2,n-2] so n=p+pc.
I did a test for the first 2000 even numbers (my computer slows down very much after that point) and prepared a graph showing ${\pi(n)}$, ${\pi(n/2)}$, G${\pi(n/2)}$ and the linear interpolation of the value of G${\pi(n/2)}$ (a kind of average value of the growing G${\pi}$ set of primes.
Then I played with lower values of the ${\pi}$ function looking for a lower bound of the linear interpolation. Finally ${\pi(n/10)}$ seemed a good value to define a lower bound for G${\pi(n/2)}$ because the linear interpolation is always over that value (at least in the test it seems so).


Here is the graph.



When I checked the results, I wondered if there is a way to work on the conjecture through a relationship between Goldbach's comet value for n, and ${\pi(n)}$, specially if the density of the subset of primes in [2,n/2] that are symmetrical in n/2 is always greater than the density of primes in a lower subset of ${\pi}$ function (e.g. ${\pi(n/10)}$).

So the question is: is there any relationship already known or being researched with ${\pi(n)}$ or only the extended Goldbach conjecture R(n) function is the correct approach to a solution of the Goldbach conjecture?
I don't believe there's any deep relationship between $\pi(n)$ and your function $R(n).$ For one thing, their expected 'bulk' rates of growth are different: $R(n)$ should grow something like $n/\log^2(n),$ while $\pi(n)\sim n/\log(n).$ So dividing $\pi(n)$ by 10 isn't going to be enough -- you'll need to divide by more and more as $n$ grows. For example, $\pi(10^{14})=3204941750802$ but $R(10^{14})=90350630388$ and so their quotient is already 35.

Second, $\pi(n)$ grows smoothly, in the sense that $\pi(n)\le\pi(n+1)\le\pi(n)+1.$ But $R(n)$ grows wildly, more even than your graph so far suggests. Essentially, $R$ is sensitive to the small prime divisors in $n$, while $\pi$ doesn't care. So $R(30n)$ grows differently from $R(30n+1)$, while the same is not true for $\pi(30n)$ and $\pi(30n+1).$
There is a relationship between Goldbach's comet G(n) and the prime counting function π(n). This sum gives the exact result for G(n) (A002372)
$$G(n) = \sum_{i=2}^{\pi(2n)} \pi(2n-p(i))-\pi(2n-1-p(i))$$ where p(n) is the nth prime.