On a Mertens-type conjecture for number fields

TL;DR

This paper introduces a number field analogue of the Mertens conjecture, proves its falsity for most number fields, and studies the distribution and properties of the generalized Mertens function in specific cases.

Contribution

It formulates a new Mertens-type conjecture for number fields, proves its general falsity, and analyzes the distribution and properties of the generalized Mertens function.

Findings

Falsity of the conjecture for all but finitely many number fields of any degree

Existence of a logarithmic limiting distribution for the Mertens function

Properties of the generalized Mertens function in dicyclic 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.