  Let $\sigma(n)$ 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 $\sigma(n)/n
\ge t$. We give an improved asymptotic result for $\log A(t)$ as $t$ grows
unbounded. The same result holds if $\sigma(n)/n$ is replaced by $n/\phi(n)$,
where $\phi(n)$ is Euler's totient function.
