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The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect

Masahiro Mine

Abstract

Arising from the factorizations of Dedekind zeta-functions of cubic fields, we obtain Artin $L$-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin $L$-functions for families of non-Galois cubic fields in conductor aspect. We prove that various mean values of the Artin $L$-functions are represented by integrals involving a density function which can be explicitly constructed. By the class number formula, the result is applied to the study on the distribution of class numbers of cubic fields.

The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect

Abstract

Arising from the factorizations of Dedekind zeta-functions of cubic fields, we obtain Artin -functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin -functions for families of non-Galois cubic fields in conductor aspect. We prove that various mean values of the Artin -functions are represented by integrals involving a density function which can be explicitly constructed. By the class number formula, the result is applied to the study on the distribution of class numbers of cubic fields.

Paper Structure

This paper contains 27 sections, 44 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Artin $L$-functions associated with cubic fields
  3. Related topics
  4. Statement of results
  5. Preliminaries
  6. Properties of the Artin $L$-function
  7. The $z$-th divisor function
  8. Results from probability theory
  9. Other preliminary lemmas
  10. The density functions $C_\sigma$ and $K_\sigma$
  11. Random Euler products
  12. Construction of the density functions
  13. Analytic properties of the density functions
  14. Proof of Theorem \ref{['thm:2.1']} $(\mathrm{i})$
  15. Proof of Theorem \ref{['thm:2.1']} $(\mathrm{ii})$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\sigma>7/8$ be a real number. Then there exists a non-negative $C^\infty$-function $C_\sigma$ on $\mathbb{R}$ such that holds for any $a \in\mathbb{R}$. Furthermore, the Fourier transform of $C_\sigma$ is represented as where $p$ runs through all prime numbers.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Ihara--Matsumoto IharaMatsumoto2011b
  • Theorem 1.4: Cho--Kim ChoKim2018
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 2.6
  • ...and 58 more