On A Divisor Sum Involving Pairwise Relatively Prime Positive Integers

TL;DR

This paper investigates the count of pairwise relatively prime positive integer tuples with bounded product, introduces a generalized divisor sum involving a Möbius function variant, and derives related identities and auxiliary functions.

Contribution

It presents a new divisor sum formulation using a generalized Möbius function and establishes identities involving square-free numbers and auxiliary number theoretic functions.

Findings

Derived identities for the divisor sum involving pairwise coprime tuples.

Formulated a weighted sum of reciprocals of square-free numbers up to n.

Introduced auxiliary functions to analyze the divisor sum.

Abstract

The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and establish some identities. One such identity is a weighted sum of reciprocal of square-free numbers not exceeding $n$. Some auxiliary number theoretic functions are introduced to formulate this sum.