Positive rational number of the form $\phi (k{m}^a)/\phi (l{n}^b)$

Abstract

Let $k,l,a$ and $b$ be positive integers with $\max \{a,b\}\ge 2$. In this paper, we show that every positive rational number can be written as the form $\phi (k{m}^a)/\phi (l{n}^b)$, where $m,n\in \mathbb{N}$ if and only if $\gcd (a,b)=1$ or $(a,b,k,l)=(2,2,1,1)$. Moreover, if $\gcd (a,b)>1$, then the proper representation of such representation is unique.