A mean value of the representation function for the sum of two primes in arithmetic progressions

TL;DR

This paper derives an asymptotic formula for the average number of representations of integers as sums of two primes in arithmetic progressions, assuming a variant of the Generalized Riemann Hypothesis that allows real zeros.

Contribution

It improves previous results by providing a more general asymptotic formula under a broader hypothesis, extending work on the Goldbach problem in arithmetic progressions.

Findings

Established an asymptotic formula for the mean value of the representation function.

Extended previous results to a broader class of hypotheses.

Improved understanding of prime sums in arithmetic progressions.

Abstract

In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. R\"uppel in 2009, and the generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.