Extended Goldbach conjecture with Hardy–Littlewood asymptotic

Let $k$ be a positive integer and let $C_{2,k}$ be the prime-pair constant. Extended Goldbach conjecture. The number of ways of writing $2k$ as a sum of two primes is asymptotic to

$$2C_{2,k}\int_2^N\frac{dx}{(\log x)^2}\sim\frac{2C_{2,k}N}{(\log N)^2}.$$

This is the Hardy–Littlewood heuristic for representations of an even number as a sum of two primes; it remains unproved in the stated asymptotic form.