Define the Möbius function $\mu$, and explain what it means for it to be multiplicative.

Show that for every positive integer $n$

$$\sum_{d \mid n} \frac{\mu(d)^{2}}{\phi(d)}=\frac{n}{\phi(n)}$$

where $\phi$ is the Euler totient function.

Fix an integer $k \geqslant 1$. Use the Chinese remainder theorem to show that there are infinitely many positive integers $n$ for which

$$\mu(n)=\mu(n+1)=\cdots=\mu(n+k)$$