The distribution functions of $\sigma (n)/n$ and $n/\varphi (n)$

Abstract:

Let $\sigma (n)$ be the sum of the positive divisors of $n$. We show that the natural density of the set of integers $n$ satisfying $\sigma (n)/n\ge t$ is given by $\exp \left \{ -e^{t e^{-\gamma }} \left (1+O\left ({t^{-2}}\right )\right ) \right \}$, where $\gamma$ denotes Euler’s constant. The same result holds when $\sigma (n)/n$ is replaced by $n/\varphi (n)$, where $\varphi$ is Euler’s totient function.