Weak Hardy–Littlewood conjecture for the Goldbach representation function

Let $R(n)$ denote the number of representations of an integer $n$ as a sum of two primes. Weak Hardy–Littlewood conjecture. There exists an absolute constant $c_3>0$ such that, for every even integer $n\geq 4$,

$$R(n)\geq \frac{c_3n}{\log^2 n}.$$

This lower-bound hypothesis is weaker than the full Hardy–Littlewood asymptotic formula and is used in the source to obtain improved bounds for Siegel zeros; its general validity remains open.