On Mobius and Liouville functions of order $k$

TL;DR

This paper introduces Mobius and Liouville functions of order k in a number field, providing formulas for their partial sums and analyzing the distribution of k-free ideals using elementary number theory and complex analysis.

Contribution

It defines new functions of order k in number fields and derives formulas for their partial sums, extending classical concepts in algebraic number theory.

Findings

Formulas for partial sums of Mobius and Liouville functions of order k

Analysis of the distribution of k-free ideals in number fields

Extension of classical functions to higher order in algebraic number theory

Abstract

Let $F$ be a number field, $k$ a positive integer. In this paper, we define the Mobius and Liouville functions of order $k$ in $F$. We give a formula about the partial sums of them by using elementary number theory and complex analysis. Moreover, we also consider the number of $k$-free ideals of the integer ring of $F$.