Question 72
For a positive integer $n \ge 2$ let $\tau(n)$ denote the number of positive divisors of $n$, and let $\sigma(n)$ be the sum of these divisors. Prove that
$$
\frac{n+1}{2} \ge \frac{\sigma(n)}{\tau(n)} \ge \frac{3}{2\sqrt{2}}\sqrt{n}.
$$
For each of these inequalities also determine when the equality holds.
$$
\frac{n+1}{2} \ge \frac{\sigma(n)}{\tau(n)} \ge \frac{3}{2\sqrt{2}}\sqrt{n}.
$$
For each of these inequalities also determine when the equality holds.