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

TL;DR

This paper improves the asymptotic understanding of the distribution of the ratios   n/\u03c6(n) and    (n)/n, providing more precise density estimates for large values.

Contribution

It offers an improved asymptotic formula for the natural density of integers with large    (n)/n and n/(n), extending previous results.

Findings

Enhanced asymptotic estimates for    (n)/n and n/(n) densities.

Asymptotic behavior holds for both    (n)/n and n/(n).

Provides a deeper understanding of the distribution of divisor sum ratios.

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