The binary Goldbach conjecture with primes in arithmetic progressions with large modulus
Volume 159 / 2013
Acta Arithmetica 159 (2013), 227-243
MSC: 11F32, 11F25.
DOI: 10.4064/aa159-3-2
Abstract
It is proved that for almost all prime numbers $k\leq N^{1/4-\epsilon},$ any fixed integer $b_{2}$, $(b_{2},k)=1,$ and almost all integers $b_{1}$, $1\leq b_{1}\leq k$, $(b_{1},k)=1, $ almost all integers $n$ satisfying $n\equiv b_{1}+b_{2}\,\, ({\rm mod}\,\, k)$ can be written as the sum of two primes $p_{1}$ and $p_{2}$ satisfying $p_{i}\equiv b_{i}\,\,({\rm mod}\,\, k)$, $i=1,2.$ For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.
