Average orders of Goldbach Estimates in Arithmetic Progressions

TL;DR

This paper investigates the average number of Goldbach representations of integers as sums of two primes in various arithmetic progressions, providing asymptotic results and demonstrating their optimality through omega-results.

Contribution

It presents new asymptotic formulas for Goldbach representations in arithmetic progressions and proves their optimality with omega-results.

Findings

Asymptotic formulas for average Goldbach representations

Omega-results showing the bounds are essentially best possible

Enhanced understanding of Goldbach representations in arithmetic progressions

Abstract

We obtain asymptotic results on the average numbers of Goldbach representations of an interger as the sum of two primes in different arithmetic progressions. We also prove an omega-result showing that the asymptotic result is essentially the best possible.