The Goldbach Conjecture With Summands In Arithmetic Progressions

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