Ces\`aro averages for Goldbach representations with summands in arithmetic progressions

TL;DR

This paper studies the average number of ways to express even numbers as sums of two primes within specific arithmetic progressions, providing uniform results across a range of moduli.

Contribution

It introduces a uniform analysis of Goldbach representations with primes in arithmetic progressions, extending previous results to a broader range of moduli.

Findings

Provides uniform bounds for weighted averages of Goldbach representations

Extends understanding of prime sums in arithmetic progressions

Offers new insights into the distribution of prime sums across moduli

Abstract

We consider weighted averages of the number of representations of an even integer as a sum of two prime numbers, where each summand lies in a given arithmetic progression modulo a common integer $q$. Our result is uniform in a suitable range for $q$.