# The Average Number of Goldbach Representations and Zero-Free Regions of the Riemann Zeta-Function

Source: [The Average Number of Goldbach Representations and Zero-Free Regions of the Riemann Zeta-Function](<https://www.sota2.com/research/paper/68fa5d561e209131ce9d9626>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T18:12:08.590Z

Authors: Keith Billington, Maddie Cheng, Jordan Schettler, Ade Irma Suriajaya

Published: 2023-06-15T12:55:00.000Z

## Abstract

In this paper, we prove an unconditional form of Fujii's formula for the average number of Goldbach representations and show that the error in this formula is determined by a general zero-free region of the Riemann zeta-function, and vice versa. In particular, we describe the error in the unconditional formula in terms of the remainder in the Prime Number Theorem which connects the error to zero-free regions of the Riemann zeta-function.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2306.09102>)

[PDF](<https://arxiv.org/pdf/2306.09102.pdf>)