Average orders of Goldbach representations in Arithmetic Progressions

The Goldbach conjecture is one of the oldest open problems today. In the original problem, we already know some results on the average orders of Goldbach representations, e.g., the asymptotic formula, omega-results, and the relation to the Riemann Hypothesis. We now consider this problem in arithmetic progressions. Some results are known under special conditions (Generalized Riemann Hypothesis with real zeros) or in arithmetic progressions with a common modulus. In this talk, we will prove an asymptotic formula and an omega-result on the average orders of Goldbach representations of an integer as the sum of two primes in different arithmetic progressions.