At the last line of the proof:
$\lambda^{-1}(n)=\mu(n)\lambda(n)=\mu^2(n)=|\mu(n)|$.
Why $\mu(n)\lambda(n)=\mu^2(n)$? How to prove this?
At the last line of the proof:
$\lambda^{-1}(n)=\mu(n)\lambda(n)=\mu^2(n)=|\mu(n)|$.
Why $\mu(n)\lambda(n)=\mu^2(n)$? How to prove this?
If $n$ is not squarefree, both sides are zero.
If $n$ is squarefree, then $\mu(n)=\lambda(n)$.