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

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

2010-11-18

arxiv.org/abs/1011.4262v1

Mathematics

Number Theory

11 pages

Scientific paper

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.

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