Hello,

This question is a follow-up from About Goldbach's conjecture.

As $N_{2}(n)=\sum_{r\leq n}1_{\mathbb{P}}(n-r)1_{\mathbb{P}}(n+r)$, Chebotarev's theorem allows to write:

$$\dfrac{N_{2}(n)}{\pi(n)}\sim \dfrac{N_{1}(n)}{\varphi(P_{ord_C(n)})}$$

So that

$$N_2(n)\sim \pi(n) \dfrac{N_{1}(n)}{\prod_{2\lt p\le \sqrt{2n-3}}(p-1)}$$.

One can easily show that $$N_{1}(n)={\prod_{2\lt p\le \sqrt{2n-3}}(p-2)}\prod_{p\vert n \\ p>2}\dfrac{p-1}{p-2}$$.

Thus one gets

$$N_{2}(n)\sim \pi(n)\dfrac{C}{\log\sqrt{2n-3}}\prod_{p\vert n \\ p>2}\dfrac{p-1}{p-2}$$ and, through the prime number theorem:

$$N_{2}(n)\sim \dfrac{Kn}{\log^{2} n}\prod_{p\vert n \\ p>2}\dfrac{p-1}{p-2}$$ with $K>0$.

As $N_{2}(n)$ is the number $G(2n)$ of couples $(p,q)$ such that $p+q=2n$, $p\leq q$, $p$ and $q$ primes, this shows that

$$G(2n)\sim \dfrac{Kn}{\log^{2} n}\prod_{p\vert n \\ p>2}\dfrac{p-1}{p-2}$$.

I have been told on a French maths forum that Hardy and Littlewood rigorously proved that if such a $K$ exists, then its value is such that their conjecture sometimes knowns as "extended Goldbach's conjecture" is true. Would you have a reference where this proof is given?