Hello,


This question is a follow-up from [About Goldbach's conjecture](https://mathoverflow.net/questions/61842/about-goldbachs-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?  

Thanks in advance.



About Extended Goldbach's conjecture
This is from Section 4 of Hardy and Littlewood's "[On some problems of partitio numerorum III: On the expression of a number as a sum of primes](http://www.springerlink.com/content/d700t434065k0230/?MUD=MP)" 1923 Acta Math. 44: 1–70.


