Explicit estimates for the Goldbach summatory function

Abstract

In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the average order of its summatory function both in the classical case and in arithmetic progressions. In addition, we derive new explicit estimates for sums over zeros and for the function $\psi(u,\chi)$. Our results are general and describe how the explicit bounds depend on other known explicit estimates. These support the known asymptotic results under the (Generalised) Riemann Hypothesis involving error terms.