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On a Mertens-type conjecture for number fields

Daniel Hu, Ikuya Kaneko, Spencer Martin, Carl Schildkraut

Abstract

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalized Mertens function of certain dicyclic number fields as consequences of Artin factorization.

On a Mertens-type conjecture for number fields

Abstract

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalized Mertens function of certain dicyclic number fields as consequences of Artin factorization.

Paper Structure

This paper contains 24 sections, 40 theorems, 180 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The naïve Mertens-type conjecture over a number field
  3. Logarithmic limiting distributions
  4. Preliminaries
  5. Notation and conventions
  6. Analytic properties of Dedekind zeta functions
  7. $\Omega$-type lemmas for $M_K(x)$ and disproofs of the naïve Mertens-type conjecture in degenerate cases
  8. The explicit formula for $M_K(x)$
  9. Unconditional failure of the naïve Mertens-type conjecture
  10. The general result
  11. Imaginary quadratic fields
  12. Real quadratic fields
  13. General number fields
  14. Proof of Theorem \ref{['thm:limiting-distribution-abelian']}
  15. Support for the conjecture \ref{['eq:gonek-hejhal-nfield']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

We have the following statements:

Theorems & Definitions (74)

  • Conjecture : The naïve Mertens-type conjecture over $K$
  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6: Linear independence conjecture for $\zeta_K(s)$
  • Theorem 1.7
  • Definition 1.8
  • ...and 64 more