Question 84
Given $n \in \mathbb{N}$, let $\sigma(n)$ denote the sum of the divisors of $n$ and $\varphi(n)$ denote the number of positive integers $m \leq n$ for which $\operatorname{gcd}(m, n)=1$. Show that for all $n \in \mathbb{N}$,
$$
\frac{1}{\varphi(n)}+\frac{1}{\sigma(n)} \geq \frac{2}{n}
$$
and determine when equality holds.
$$
\frac{1}{\varphi(n)}+\frac{1}{\sigma(n)} \geq \frac{2}{n}
$$
and determine when equality holds.