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Average orders of Goldbach Estimates in Arithmetic Progressions

Thi Thu Nguyen

TL;DR

This work analyzes the average number of weighted Goldbach representations of integers as sums of two primes in distinct arithmetic progressions, without assuming GRH. Using the circle method and Dirichlet L-functions, it derives an unconditional asymptotic formula for $S(X,q_1,q_2,a_1,a_2)$ with a main term $X^2/(2\varphi(q_1)\varphi(q_2))$ and explicit secondary contributions from $H(X,q_i,a_i)$ and a Siegel-zero term, plus a quantified error. It also proves an omega-result showing the asymptotic is essentially best possible in general, and it provides sharp second-moment bounds for the Chebyshev function in arithmetic progressions to control the error. The results improve upon prior work (e.g., BHMS) and extend Goldbach-type averages to the case of different moduli, contributing to a deeper understanding of prime sums in APs and their irregularities due to exceptional zeros.

Abstract

We obtain asymptotic results on the average numbers of Goldbach representations of an interger as the sum of two primes in different arithmetic progressions. We also prove an omega-result showing that the asymptotic result is essentially the best possible.

Average orders of Goldbach Estimates in Arithmetic Progressions

TL;DR

This work analyzes the average number of weighted Goldbach representations of integers as sums of two primes in distinct arithmetic progressions, without assuming GRH. Using the circle method and Dirichlet L-functions, it derives an unconditional asymptotic formula for with a main term and explicit secondary contributions from and a Siegel-zero term, plus a quantified error. It also proves an omega-result showing the asymptotic is essentially best possible in general, and it provides sharp second-moment bounds for the Chebyshev function in arithmetic progressions to control the error. The results improve upon prior work (e.g., BHMS) and extend Goldbach-type averages to the case of different moduli, contributing to a deeper understanding of prime sums in APs and their irregularities due to exceptional zeros.

Abstract

We obtain asymptotic results on the average numbers of Goldbach representations of an interger as the sum of two primes in different arithmetic progressions. We also prove an omega-result showing that the asymptotic result is essentially the best possible.
Paper Structure (5 sections, 18 theorems, 136 equations)

This paper contains 5 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Some calculations for the proof of Theorem \ref{['t1']}
  3. Second moments of Chebyshev function in arithmetic progressions
  4. Proof of the part (A) Theorem \ref{['t1']}
  5. Proof of the part (B) Theorem \ref{['t1']}

Key Result

Theorem 1.1

Fujii Suppose that the Riemann Hypothesis (RH) is true. Then we have where $H(X)=-2\sum_\rho \dfrac{X^{\rho+1}}{\rho(\rho+1)}$ is the sum over the non-trivial zeros $\rho$ of the Riemann zeta function counted with multiplicity and $E(X) \ll (X\log X)^{4/3}$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • ...and 23 more