Dirichlet Convolution involving mobius function I want to show the sum
$$\sum_{d/n}|\mu(d)|\mu(\frac{n}{d})$$
only has a result not $0$ when $n$ is a perfect square (and then it's $\mu(\sqrt n)$. I think it might have something to do with perfect squares having an odd number of divisors because $d=n/d$, but I don't know how to use that to prove it?

Multiplicative arithmetic functions ($\gcd(a,b)=1\Rightarrow f(ab)=f(a)f(b)$) are closed under the operation of Dirichlet convolution. In particular $\mu$ and $|\mu|$ are multiplicative. If $n=p^k$ then
$$\sum_{\large d\mid n}|\mu(d)|\mu(n/d)=\sum_{e=0}^k |\mu(p^e)|\mu(p^{k-e})=\begin{cases} 0  & k\ge3 \\ -1 & k=2 \\ 0 & k=1 \\ +1 & k=0\end{cases}$$
The desired conclusion follows.
