Paper detail

The distribution functions of $σ(n)/n$ and $n/ϕ(n)$, II

Let $σ(n)$ be the sum of the positive divisors of $n$, and let $A(t)$ be the natural density of the set of positive integers $n$ satisfying $σ(n)/n \ge t$. We give an improved asymptotic result for $\log A(t)$ as $t$ grows unbounded. The same result holds if $σ(n)/n$ is replaced by $n/ϕ(n)$, where $ϕ(n)$ is Euler's totient function.