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The average number of Goldbach representations over multiples of $q$

Karin Ikeda, Ade Irma Suriajaya

TL;DR

This work analyzes the average number of Goldbach representations for integers that are multiples of $q$, quantified by $G_q(N)$, under the generalized Riemann hypothesis. Building on the circle-method approach and generating-function framework of prior work, it expresses $G_q(N)$ in terms of Dirichlet-character sums and analyzes main and error terms via a decomposition into $\mathcal{I}_1$, $\mathcal{I}_2$, and $\mathcal{I}_3$. The principal-character contribution yields the expected main term $\frac{G(N)}{\phi(q)}$, while Gallagher's lemma and GRH-based bounds control the off-diagonal/character terms, resulting in the sharp bound $G_q(N)=\frac{G(N)}{\phi(q)}+\mathcal{O}(N\log^3 N)$. The approach extends and refines the Languasco–Zaccagnini and GS methodology to achieve a relatively tight error with $C=3$, contributing to a clearer understanding of the distribution of Goldbach representations within arithmetic progressions under GRH.

Abstract

We discuss the evaluation of the average number of Goldbach representations for integers which are multiples of $q$ introduced by Granville. We improve an estimate given by Granville under the generalized Riemann hypothesis.

The average number of Goldbach representations over multiples of $q$

TL;DR

This work analyzes the average number of Goldbach representations for integers that are multiples of , quantified by , under the generalized Riemann hypothesis. Building on the circle-method approach and generating-function framework of prior work, it expresses in terms of Dirichlet-character sums and analyzes main and error terms via a decomposition into , , and . The principal-character contribution yields the expected main term , while Gallagher's lemma and GRH-based bounds control the off-diagonal/character terms, resulting in the sharp bound . The approach extends and refines the Languasco–Zaccagnini and GS methodology to achieve a relatively tight error with , contributing to a clearer understanding of the distribution of Goldbach representations within arithmetic progressions under GRH.

Abstract

We discuss the evaluation of the average number of Goldbach representations for integers which are multiples of introduced by Granville. We improve an estimate given by Granville under the generalized Riemann hypothesis.
Paper Structure (2 sections, 8 theorems, 70 equations)

This paper contains 2 sections, 8 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Proof of \ref{['main']}

Key Result

Theorem 1.1

F Assuming the Riemann hypothesis, we have where the sum is over the nontrivial zeros $\rho$ of the Riemann zeta function, and the Riemann hypothesis states that ${\rm Re}(\rho)=1/2$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Gallagher
  • Lemma 2.4
  • proof