Density properties of fractions with Euler's totient function

TL;DR

This paper investigates the density of fractions involving Euler's totient function, establishing conditions for density in certain intervals, and provides algorithms and open questions related to these properties.

Contribution

It proves density results for fractions with Euler's totient function, characterizes when isolated points occur, and introduces an algorithm and open problems related to these fractions.

Findings

Fractions of the form φ(an+b)/(cn+d) are dense in a specific interval.

Complete characterization of when isolated fractions occur outside the interval.

An algorithm to count n with rad(an+b) dividing g, and an open question generalizing Arnold's problem.

Abstract

We prove that for all constants $a\in\N$, $b\in\Z$, $c,d\in\R$, $c\neq 0$, the fractions $\phi(an+b)/(cn+d)$ lie dense in the interval $]0,D]$ (respectively $[D,0[$ if $c<0$), where $D=a\phi(\gcd(a,b))/(c\gcd(a,b))$. This interval is the largest possible, since it may happen that isolated fractions lie outside of the interval: we prove a complete determination of the case where this happens, which yields an algorithm that calculates the amount of $n$ such that $\rad(an+b)|g$ for coprime $a,b$ and any $g$. Furthermore, this leads to an interesting open question which is a generalization of a famous problem raised by V.~Arnold. For the fractions $\phi(an+b)/\phi(cn+d)$ with constants $a,c\in\N,b,d\in\Z$, we prove that they lie dense in $]0,\infty[$ exactly if $ad\neq bc$.