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

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.