The Goldbach conjecture with summands in arithmetic progressions

TL;DR

This paper proves that almost all even numbers up to N can be expressed as the sum of two primes in specified arithmetic progressions for a wider range of moduli r, improving previous bounds from N^{1/3} to N^{1/2}.

Contribution

It extends the range of moduli r for which Goldbach-type representations in arithmetic progressions hold, advancing the understanding of prime sums in modular settings.

Findings

Almost all even numbers up to N can be expressed as sums of two primes in specified progressions for r up to N^{1/2}.

Improved bounds from previous N^{1/3} range to N^{1/2} for the modulus r.

Enhanced results on variations of the Goldbach problem in arithmetic progressions.

Abstract

We prove that, for almost all $r \leq N^{1/2}/\log^{O(1)}N$, for any given $b_1 \mod r$ with $(b_1, r) = 1$, and for almost all $b_2 \mod r$ with $(b_2, r) = 1$, we have that almost all natural numbers $2n \leq N$ with $2n \equiv b_1 + b_2 \mod r$ can be written as the sum of two prime numbers $2n = p_1 + p_2$, where $p_1 \equiv b_1 \mod r$ and $p_2 \equiv b_2 \mod r$. This improves the previous result which required $r \leq N^{1/3}/\log^{O(1)}N$ instead of $r \leq N^{1/2}/\log^{O(1)}N$. We also improve some other results concerning variations of the problem.