Density properties of fractions with Euler's totient function

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 $ad\neq bc$.