Hardy–Littlewood's binary Goldbach asymptotic conjecture

Let $r_2'(N)$ denote the number of representations of $N$ as a sum of two primes, and let $\mathfrak{S}(N)$ be the singular series

$$\mathfrak{S}(N)=\prod_p\left(1+\frac{c_p(N)}{(p-1)^2}\right)=\prod_{p\nmid N}\left(1-\frac{1}{(p-1)^2}\right)\prod_{p\mid N}\left(1+\frac{1}{p-1}\right).$$

For even integers $N$, let $r_2(N)$ denote the corresponding von Mangoldt-weighted representation count. Hardy–Littlewood's conjecture. As $N\to\infty$ through even integers,

$$r_2(N)\sim \mathfrak{S}(N)N,$$

Equivalently,

$$r_2'(N)\sim \mathfrak{S}(N)\frac{N}{(\log N)^2}.$$

This is the quantitative form of the binary Goldbach conjecture: it predicts not only that every sufficiently large even integer has a representation as the sum of two primes, but also the precise asymptotic number of such representations. The conjecture remains unresolved.