On an Average Goldbach Representation Formula of Fujii

TL;DR

This paper extends Fujii's formula for the average Goldbach representations to an unconditional form and explores applications under various assumptions about the zeros of the Riemann zeta-function.

Contribution

It provides an unconditional version of Fujii's average Goldbach representation formula and investigates its applications under different zero hypotheses.

Findings

Unconditional formula for average Goldbach representations

Applications conditional on zero conjectures

Enhanced understanding of Goldbach representations

Abstract

Fujii obtained a formula for the average number of Goldbach representations with lower order terms expressed as a sum over the zeros of the Riemann zeta-function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result, and obtain applications conditional on various conjectures on zeros of the Riemann zeta-function.