The Goldbach conjecture with summands in arithmetic progressions

We prove that, for almost all $r \leq N^{1/2}/\log^{O(1)}N$, for any given $b_1 \mod r$ with $(b_1, r) = 1$, and for almost all $b_2 \mod r$ with $(b_2, r) = 1$, we have that almost all natural numbers $2n \leq N$ with $2n \equiv b_1 + b_2 \mod r$ can be written as the sum of two prime numbers $2n = p_1 + p_2$, where $p_1 \equiv b_1 \mod r$ and $p_2 \equiv b_2 \mod r$. This improves the previous result which required $r \leq N^{1/3}/\log^{O(1)}N$ instead of $r \leq N^{1/2}/\log^{O(1)}N$. We also improve some other results concerning variations of the problem.