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

Abstract

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 $n/\varphi (n)$, where $\varphi (n)$ is Euler's totient function.

11 pages