Doubt in proving inverse of Liouville's function $\lambda(n) $

Question

I have following doubt from Apostol s Introduction to analytic number theory regarding proving inverse of Liouville's function.

It's image -

My doubt is in last line of proof - how does Apostol writes $\mu (n) $ ×$\lambda (n) $ = $\mu^2(n) $ ie how $\lambda (n) $ = $\mu ( n) $ ?

Can someone please explain vital

Answer 1

This follows from the definition of the Mobius function. If $n$ is square free; i.e if $n=p_1p_2...p_k$, where $p_i's$ are distinct prime factors, then $\lambda(n)=\mu(n)=\mu^2(n)(-1)^{\omega(n)}$.

Note that, $\mu(n)=1$, when $n$ is a square-free + integer with even no. of prime factors,$\mu(n)=-1$, when $n$ is a square-free + integer with odd no. of prime factors & 0 otherwise.