On the ternary Goldbach problem with primes in independent arithmetic progressions

TL;DR

This paper proves that large odd integers can be expressed as sums of three primes in independent arithmetic progressions with specified moduli and residues, extending the classical Goldbach problem.

Contribution

It establishes new results on the ternary Goldbach problem with primes in independent arithmetic progressions, under certain size and distribution conditions.

Findings

Almost all admissible residue classes are covered for large odd integers.

Existence of prime representations in specified arithmetic progressions is proven.

Results hold for a wide range of moduli with logarithmic bounds.

Abstract

We show that for every fixed $A>0$ and $\theta>0$ there is a $\vartheta=\vartheta(A,\theta)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log n)^{\theta}$. Then for all $q_{3}\leq Q_{3}$, all reduced residues $a_{3}$ mod $q_{3}$, almost all $q_{2}\leq Q_{2}$, all admissible residues $a_{2}$ mod $q_{2}$, almost all $q_{1}\leq Q_{1}$ and all admissible residues $a_{1}$ mod $q_{1}$, there exists a representation $n=p_{1}+p_{2}+p_{3}$ with primes $p_{i}\equiv a_{i} (q_{i})$, $i=1,2,3$.