Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

Abstract

Set $${T=N^{\frac{1}{3}-\epsilon}}$$ . It is proved that for all but $${\ll TL^{-H},\,H > 0}$$ , exceptional prime numbers $${k\leq T}$$ and almost all integers b 1, b 2 co-prime to k, almost all integers $${n\sim N}$$ satisfying $${n\equiv b_{1}+b_{2}(mod\,k)}$$ can be written as the sum of two primes p 1 and p 2 satisfying $${p_{i}\equiv b_{i}(mod\,k),\,i=1,2}$$ . For prime numbers $${k\leq N^{\frac{5}{24}-\epsilon}}$$ , this result is even true for all but $${\ll (\log\,N)^{D}}$$ primes k and all integers b 1, b 2 co-prime to k.