On the representation of rational numbers via Euler's totient function

Abstract

Let $b>1$ be an odd positive integer and $k,l\in \mathbb{N}$. In this paper, we show that every positive rational number can be written as $\ddot{I}†(m^2)/(\ddot{I}†(n^2){)}^b$ and $\ddot{I}†(k(m^2-1))/\ddot{I}†(l{n}^2)$, where $m,n\in \mathbb{N}$ and $\ddot{I}†$ is the Euler's totient function. At the end, some further results are discussed.