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

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.

22 pages (content in 20 pages), a student project conducted at SJSU under Kyushu University SENTAN-Q

References in corpus (3)

• On an Average Goldbach Representation Formula of Fujii
• The Prime Number Theorem and Pair Correlation of Zeros of the Riemann Zeta-Function
• On a smoothed average of the number of Goldbach representations

Cited by in corpus (2)

• $M$-functions and screw functions originating from Goldbach's problem and zeros of the Riemann zeta function
• The average number of Goldbach representations over multiples of $q$