On the number of representations in the ternary Goldbach problem with one prime number in a given residue class

ForJ3(n)=∑p1+p2+p3=np1≡a1(modq1)logp1logp2logp3,it is shown that∑q1⩽nθmax(a1,q1)=1J3(n)-n22ϕ(q1)S3(n)≪An2(logn)-Afor any A   and any θ<1/2θ<1/2, what improves a work of Tolev; S3(n)S3(n) is the corresponding singular series. A special form of a sieve of Montgomery is used.