A Linear Division-Based Recursion with Number Theoretic Applications

TL;DR

This paper introduces a linear division-based recursion and explores its applications in number theory, specifically relating to Euler's phi-function and the density of square-free numbers.

Contribution

It presents a novel recursion approach and connects it to classical and recent number theoretic results.

Findings

Established a new recursion method for number theory

Linked the recursion to Euler's phi-function theorem

Applied the recursion to analyze the density of square-free numbers

Abstract

A simple remark on infinite series is presented. This applies to a particular recursion scenario, which in turn has applications related to a classical theorem on Euler's phi-function and to recent work by Ron Brown on natural density of square-free numbers.