# Goldbach Representations in Arithmetic Progressions and zeros of Dirichlet L-functions

## TL;DR

This paper explores the relationship between representations of integers as sums of two primes in arithmetic progressions and the zeros of Dirichlet L-functions, assuming certain conjectures, and derives implications for zero distribution and error terms.

## Contribution

It establishes asymptotic formulas for prime sums in arithmetic progressions under zero assumptions and links error terms to zero locations, advancing understanding of L-function zeros.

## Key findings

- Asymptotic results on prime representations in progressions
- Connections between error terms and zeros of L-functions
- Implications for Siegel zeros and zero distribution

## Abstract

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.