Explicit estimates for the Goldbach summatory function

TL;DR

This paper provides explicit numerical estimates for the Goldbach summatory function and related sums over zeros, enhancing understanding of its behavior and supporting asymptotic results under the Riemann Hypothesis.

Contribution

It introduces new explicit bounds for the Goldbach summatory function and sums over zeros, extending previous results with detailed numerical estimates.

Findings

Explicit average order estimates for the Goldbach summatory function

New bounds for sums over zeros of L-functions

Explicit estimates for the function ψ(u,χ)

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.