Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the distribution of prime ideals in real quadratic fields

Stephan Baier, Sayantan Roy

Abstract

In this note, we give a summary of the article ``The distribution of prime ideals of imaginary quadratic fields'' by G. Harman, A. Kumchev and P. A. Lewis and establish analogous results for real quadratic fields based on the same method.

A note on the distribution of prime ideals in real quadratic fields

Abstract

In this note, we give a summary of the article ``The distribution of prime ideals of imaginary quadratic fields'' by G. Harman, A. Kumchev and P. A. Lewis and establish analogous results for real quadratic fields based on the same method.
Paper Structure (10 sections, 7 theorems, 83 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 83 equations, 2 figures.

Table of Contents

  1. Review of prime ideals in small regions for imaginary quadratic fields
  2. Größ encharaktere for real quadratic fields
  3. Main results
  4. Description of the Harman-Lewis-Kumchev method
  5. Reduction to mean value estimates for Dirichlet polynomials
  6. Primes represented by quadratic forms
  7. Link to quadratic fields
  8. Geometric interpretation
  9. Proof of Theorem \ref{['mainresult2']} for $Q(\xi,\eta)=\xi^2-d\eta^2$
  10. Proof of Theorem \ref{['mainresult2']} for the general case

Key Result

Theorem 1

Let $K$ be an imaginary quadratic number field and $\mathcal{C}$ an ideal class of $K$. Fix an ideal $\mathfrak{a}_0\in \mathcal{C}^{-1}$. Define $\theta_1:=0.765$ and $\theta_2:=1-\theta_1=0.235$. Then there is an $x_0>0$ such that if $x\ge x_0$, $x^{\theta_1}\le y\le x$, $\phi_0\in \mathbb{R}$ and where

Figures (2)

  • Figure 1: Conditions in (3) for $K=\mathbb{Q}(i)$
  • Figure 2: Sectors and hyperbolas for $Q(\xi,\eta)=\xi^2-d\eta^2$

Theorems & Definitions (7)

  • Theorem 1: Harman-Lewis-Kumchev
  • Theorem 2: Harman-Lewis-Kumchev
  • Theorem 3
  • Theorem 4
  • Proposition 5: Theorem 2.2 in Col2
  • Proposition 6
  • Corollary 7