Proving $\sum\limits_{n=1}^\infty\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\zeta(2s)}.$

Question

I am trying to show that $$D(|\mu|,s)=\sum_{n=1}^\infty\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\zeta(2s)}.$$ By a previous exercise I know that $D(\lambda,s)=\zeta(2s)/\zeta(s)$, where $\lambda$ is Liouville's function $\lambda(n)=(-1)^{\Omega(n)}$. But I don't see how to use this fact (I am not sure I should, but it looks like it would be useful).

Answer 1

The coefficient of the Dirichlet series is multiplicative, so you only need to compare the Euler factors.