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On the Proportion of Coprime Fractions in Number Fields

Walter Bridges, Johann Franke, Johann Christian Stumpenhusen

Abstract

In this paper we determine the asymptotic density of coprime fractions in those of the reduced fractions of number fields. When ordered by norms of denominators, we count a fraction as soon as it ``appears'' for the first time and no later. The natural density of coprime fractions in the set of reduced fractions may then be computed using well-known facts about Hecke $L$-functions. Furthermore, we draw some connections to the modular group and Heegner points.

On the Proportion of Coprime Fractions in Number Fields

Abstract

In this paper we determine the asymptotic density of coprime fractions in those of the reduced fractions of number fields. When ordered by norms of denominators, we count a fraction as soon as it ``appears'' for the first time and no later. The natural density of coprime fractions in the set of reduced fractions may then be computed using well-known facts about Hecke -functions. Furthermore, we draw some connections to the modular group and Heegner points.
Paper Structure (13 sections, 16 theorems, 53 equations)

This paper contains 13 sections, 16 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Extending the notion of density
  4. Main result.
  5. Density theory
  6. Density in ordered countable sets
  7. Density in $\mathbb{Q}$
  8. Density in a number field
  9. Ideal theory and reduced fractions
  10. Proof of Theorem \ref{['theo:Densityestimate']}
  11. Further results
  12. Some more ideal theory
  13. Quadratic fields

Key Result

Theorem 1.3

The natural density of coprime fractions in a number field $K$ with respect to a set of inseverable representatives $\mathcal{R}_K$ of its class group equals $\frac{1}{\sum_{\mathfrak{g} \in \mathcal{R}_K} \frac{1}{(N\mathfrak{g})^2}}$.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 2.1: see Tenenbaum on p. 350
  • Definition 2.2
  • Proposition 2.3
  • Example
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 26 more