On the irregularity of the distribution of the sums of pairs of odd primes (Q1608186)

Let \(P_2(n)\) denote the number of ways of writing \(n\) as a sum of two odd primes. Hardy and Littlewood conjectured the following quantitative version of Goldbach's conjecture: \[ P_2(n) \sim 2 \prod_{p>2} \left(1- \frac{1}{(p-1)^2}\right) \prod_{p \mid n, p\neq 2} \left( 1+\frac{1}{p-2}\right) \frac{n}{\log^2 n}. \] The author proves that this holds in the following averaged sense. Let \(2 \leq a \leq m\) be even integers. Let \(C_m= \prod_{p\mid m} \left(1- \frac{1}{(p-1)^2}\right)\). Then as \(x \rightarrow \infty\): \[ \frac{m}{x} \sum_{\overset n \leq {x} {n \equiv a \bmod m}} \frac{P_2(n)}{C_mn/\log^2 n} \sim \prod_{p \mid (a,m), p\neq 2} \left( 1+\frac{1}{p-2}\right). \]