ELMO SL 2010 N1

For a positive integer $n$ , let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$ . Prove that for all $n$ we have $$\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n},$$ and determine when equality holds.

Wenyu Cao.