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On Diophantine properties for values of Dedekind zeta functions

Jerson Caro, Fabien Pazuki, Riccardo Pengo

TL;DR

The article investigates when the Dedekind zeta-height $h_s([K]) = |\zeta_K^*(s)|$ has Northcott or Bogomolov properties across real and complex values of $s$. It establishes Bogomolov failure for all $\sigma \ge \tfrac{1}{2}$ using a combination of Soundararajan’s resonance method and random Euler product models, alongside explicit field-construction arguments to treat $\sigma>1$. For $\mathrm{Re}(s)<0$, it proves a Northcott property for the pair $(|\zeta_K^*(s)|,[K:\mathbb{Q}])$, showing that height control combined with degree bounds yields finiteness. The work unifies resonator techniques, probabilistic models, and explicit arithmetic constructions to map the Diophantine behavior of Dedekind zeta-values across the half-plane, including the critical line and its right side.

Abstract

We study the Northcott and Bogomolov property for special values of Dedekind $ζ$-functions at real values $σ\in \mathbb{R}$. We prove, in particular, that the Bogomolov property is not satisfied when $σ\geq \frac{1}{2}$. If $σ> 1$, we produce certain families of number fields having arbitrarily large degrees, whose Dedekind $ζ$-functions $ζ_K(s)$ attain arbitrarily small values at $s = σ$. On the other hand, if $\frac{1}{2} \leq σ\leq 1$, we construct suitable families of quadratic number fields, employing either Soundararajan's resonance method, which works when $\frac{1}{2} \leq σ< 1$, or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when $\frac{1}{2} < σ\leq 1$. We complete the study by proving that the Dedekind $ζ$ function together with the degree satisfies the Northcott property for every complex $s\in{\mathbb{C}}$ such that $\mathrm{Re}(s) <0$, generalizing previous work of Généreux and Lalín.

On Diophantine properties for values of Dedekind zeta functions

TL;DR

The article investigates when the Dedekind zeta-height has Northcott or Bogomolov properties across real and complex values of . It establishes Bogomolov failure for all using a combination of Soundararajan’s resonance method and random Euler product models, alongside explicit field-construction arguments to treat . For , it proves a Northcott property for the pair , showing that height control combined with degree bounds yields finiteness. The work unifies resonator techniques, probabilistic models, and explicit arithmetic constructions to map the Diophantine behavior of Dedekind zeta-values across the half-plane, including the critical line and its right side.

Abstract

We study the Northcott and Bogomolov property for special values of Dedekind -functions at real values . We prove, in particular, that the Bogomolov property is not satisfied when . If , we produce certain families of number fields having arbitrarily large degrees, whose Dedekind -functions attain arbitrarily small values at . On the other hand, if , we construct suitable families of quadratic number fields, employing either Soundararajan's resonance method, which works when , or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when . We complete the study by proving that the Dedekind function together with the degree satisfies the Northcott property for every complex such that , generalizing previous work of Généreux and Lalín.

Paper Structure

This paper contains 18 sections, 15 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical remarks
  3. Main results
  4. Notation
  5. Numbers
  6. Asymptotics
  7. Constants
  8. Special functions
  9. Arithmetic functions
  10. Dedekind zeta functions
  11. More on constants
  12. On the critical line
  13. The right half of the critical strip
  14. Resonator method
  15. Random Euler products
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\sigma \in [\frac{1}{2},+\infty)$. The function $h_\sigma \colon \mathcal{N} \to \mathbb{R}_{\geq 0}$ defined by $h_\sigma([K]) := \lvert \zeta^\ast_K(\sigma) \rvert$ does not have the Bogomolov property. More precisely, for every $\sigma \in [\frac{1}{2},+\infty)$ and every $B \in (\inf(h_\sig

Figures (1)

  • Figure :

Theorems & Definitions (36)

  • Theorem 1.1: Main Theorem
  • Theorem 1.2: Main Theorem for $\sigma=\frac{1}{2}$
  • Remark 1.3
  • Theorem 1.4: Main Theorem for $\frac{1}{2} < \sigma < 1$
  • Remark 1.5
  • Theorem 1.6: Main Theorem for $\sigma > 1$
  • Theorem 1.7
  • Remark 1.8
  • Theorem 3.1: Soundararajan
  • Theorem 3.2
  • ...and 26 more