Average Goldbach and the Quasi-Riemann Hypothesis

TL;DR

This paper shows that certain average behaviors of the Goldbach generating function are equivalent to the Riemann Hypothesis, linking additive number theory to the distribution of zeta zeros.

Contribution

It establishes a new equivalence between average order conditions on the Goldbach function and the Riemann Hypothesis, advancing understanding of their connection.

Findings

Average order conditions imply zeros have real part less than 1

Equivalence between Goldbach asymptotics and Riemann Hypothesis

Links additive number theory to zeta zero distribution

Abstract

We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an equivalence between such asymptotics and the Riemann Hypothesis.