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

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.

Keith Billington, Maddie Cheng, Jordan Schettler, Ade Irma Suriajaya · 2023