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

TL;DR

The paper investigates the average number of Goldbach representations, encapsulated by $G(N)=\sum_{n\le N}\psi_2(n)$ with $\psi_2(n)$ counting weighted prime-sum representations, and establishes an unconditional Fujii-type expansion where the error is governed by zero-free regions of the Riemann zeta-function and the remainder $R(x)$ in the Prime Number Theorem. The authors develop a converse: under a quantified decay in $G(N)$, one derives bounds on the PNT remainder $R(x)$, using a smoothed generating function framework inspired by Bhowmik–Ruzsa. They also analyze a smooth average $F(N)=\sum_n\psi_2(n)e^{-n/N}$ via zeros of $\zeta(s)$ to obtain a sharp decay rate $F(N)=N^2+O(N^{2-\eta(\log N)})$ under zero-free hypotheses, with explicit corollaries for standard zero-free regions. Overall, the work clarifies how zero-free regions influence Goldbach-type averages and provides a cohesive set of un-smoothed and smooth-averaging tools linking additive representation counts to prime distribution.

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.

Theorems & Definitions (24)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Corollary 2
• Lemma 1
• proof : Proof of Lemma \ref{['lem1']}
• Proposition 1
• proof : Proof of Proposition \ref{['thm1-interm']}
• Proposition 2