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

Andreas Weingartner

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

Number Theory 2010-11-19 v1

Authors: Andreas Weingartner

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.

Keywords

sum-of-divisors function euler totient function natural density asymptotic distribution sigma(n)/n n/phi(n)

Cite

@article{arxiv.1011.4262,
  title  = {The distribution functions of $\sigma(n)/n$ and $n/\phi(n)$, II},
  author = {Andreas Weingartner},
  journal= {arXiv preprint arXiv:1011.4262},
  year   = {2010}
}

Comments

11 pages

Related papers

View all related →