On the ternary Goldbach problem with primes in arithmetic progressions of a common module

TL;DR

This paper demonstrates that for large odd numbers, almost all primes can be expressed as sums of three primes in specific residue classes modulo k, for most k up to a certain size, extending the understanding of the Goldbach problem.

Contribution

It establishes that for large odd n, almost all residue classes mod k can be used to represent n as a sum of three primes in those classes, for most k up to n^{1/5-epsilon}.

Findings

Almost all residue classes allow such representations.

The result holds for most k up to n^{1/5-epsilon}.

Extends Goldbach problem to primes in arithmetic progressions.

Abstract

For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.