Fengyuan Chen

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

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 $φ(m^{2})/(φ(n^{2}))^{b}$ and $φ(k(m^{2}-1))/φ(ln^{2})$, where $m, n\in \mathbb{N}$ and $φ$ is the Euler's totient function. At the end, some further results are discussed.