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

Authors

Keith Billington, San Jose State University
Maddie Cheng, San Jose State University
Jordan Schettler, San Jose State UniversityFollow
Ade Irma Suriajaya, Kyushu University

Publication Date

11-26-2024

Document Type

Article

Publication Title

International Journal of Number Theory

Volume

21

Issue

2

DOI

10.1142/S1793042125500150

First Page

289

Last Page

316

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.

Keywords

Goldbach representations, Prime Number Theorem, Riemann zeta function, zero-free region

Comments

Electronic version of an article published as International Journal of Number Theory, Volume 21, Issue 2, 2025, 289-316 https://doi.org/10.1142/S1793042125500150 © 2025 World Scientific Publishing Co Pte Ltd https://www.worldscientific.com/worldscinet/ijnt

Department

Mathematics and Statistics

Recommended Citation

Keith Billington, Maddie Cheng, Jordan Schettler, and Ade Irma Suriajaya. "The Average Number of Goldbach Representations and Zero-free Regions of the Riemann Zeta Function" International Journal of Number Theory (2024): 289-316. https://doi.org/10.1142/S1793042125500150