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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?

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up vote 3 down vote accepted

If $n$ is not squarefree, both sides are zero.

If $n$ is squarefree, then $\mu(n)=\lambda(n)$.

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