On asymptotics for the number of ways to write $2n$ as the sum of two primes

Question

Let $g(n)$ be the number of ways to write $2n$ as the sum of two primes. Define $G(n) = g(2) + \cdot \cdot \cdot + g(n)$. Are there any conjectured or known asymptotics or bounds for $G(n)$? Especifically, is it known or conjectured that $G(n) \sim \frac{n^2}{\log(n)^2}$?

Answer 1

Unconditionally we have that $$G(N)\sim\frac{2N^2}{\log^2(2N)}.$$

To see why, let $1_{\mathcal{P}}(n)$ denote the indicator function for the odd primes. Then $$G(N)=\sum_{n=1}^{2N} \sum_{a+b=n} 1_{\mathcal{P}}(a)1_{\mathcal{P}}(b).$$ Since $1_{\mathcal{P}}(m)=0$ when $m<0$, we may extending the inner sum to obtain

$$G(N)=\sum_{n=1}^{2N} \sum_{a=1}^{2N} 1_{\mathcal{P}}(a)1_{\mathcal{P}}(n-a),$$ and then by switching the order of summation

$$G(N)=\sum_{a=1}^{2N} 1_{\mathcal{P}}(a)\sum_{n=1}^{2N} 1_{\mathcal{P}}(n-a)$$

$$=\sum_{a=1}^{2N} 1_{\mathcal{P}}(a)\sum_{n=1}^{2N-a} 1_{\mathcal{P}}(n)$$

$$=\sum_{a=1}^{2N} 1_{\mathcal{P}}(a)(\pi(2N-a)-1),$$ and by using the prime number theorem, this is asymptotic to $$\frac{2N^2}{\log^2(2N)}.$$