Let $n \in 2\mathbb{N}^*$ large enough.

You search for the quantative version of Goldbach's conjecure, Hardy and Littlewood in there 1923 paper "Some problems of ‘Partitio numerorum’; III : On the expression of a number as a sum of primes", conjecture that : $$G(n) \sim 2 C_2 \displaystyle {\small \Big( \prod_{\substack{p | n \\ \text{p prime} \\ 3 \leqslant p}} {\normalsize \dfrac{p-1}{p-2}} \Big)} \dfrac{n}{\log(n)^2}.$$

Where $G(n) = \#\{(p, n-p) \in \mathbb{P}^2 \, | \, p \leqslant n\}$, and : $C_2 = \displaystyle{\small \prod_{\substack{3 \leq p \\ \text{p prime}}} \left({\normalsize 1-\dfrac{1}{(p-1)^2}}\right)}$.

This conjecture agree perfectely with numeric checks, but unfortunately not proven up to now (and no hope to prove it soon).

You can see my try here : is there a link with the probabilistic model for prime numbers?

Let $n \in 2\mathbb{N}^*$ large enough.

$$n = p+q, \ (p,q)\in\mathbb{P}^2 \iff (p, n-p) \in \mathbb{P}^2$$

You search for the quantative version of Goldbach's conjecure, Hardy and Littlewood in there 1923 paper "Some problems of ‘Partitio numerorum’; III : On the expression of a number as a sum of primes", conjecture that : $$G(n) \sim 2 C_2 \displaystyle {\small \Big( \prod_{\substack{p | n \\ \text{p prime} \\ 3 \leqslant p}} {\normalsize \dfrac{p-1}{p-2}} \Big)} \dfrac{n}{\log(n)^2}.$$

Where $G(n) = \#\{(p, n-p) \in \mathbb{P}^2 \, | \, p \leqslant n\}$, and : $C_2 = \displaystyle{\small \prod_{\substack{3 \leq p \\ \text{p prime}}} \left({\normalsize 1-\dfrac{1}{(p-1)^2}}\right)}$.

This conjecture agree perfectely with numeric checks, but unfortunately not proven up to now (and no hope to prove it soon).

You can see my try here : is there a link with the probabilistic model for prime numbers?