Orders and partitions of integers induced by arithmetic functions

Abstract

We pursue the question how integers can be ordered or partitioned according to their divisibility properties. Based on pseudometrics on $\mathbb{Z}$, we investigate induced preorders, associated equivalence relations, and quotient sets. The focus is on metrics or pseudometrics on $\mathbb{D}_n$, the set of divisors of a given modulus $n\in\mathbb{N}$, that can be extended to pseudometrics on $\mathbb{Z}$. Arithmetic functions can be used to generate such pseudometrics. We discuss several subsets of additive and multiplicative arithmetic functions and various combinations of their function values leading to binary metric functions that represent different divisibility properties of integers. We conclude this paper with numerous examples and review the most important results. As an additional result, we derive a necessary condition for the truth of the odd k-perfect number conjecture.