From this question here:
Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power representation modulo $2$, or in formula: $$ \mathrm{core}(p^e) = p^{e\mod 2}, $$ with $\mathrm{core}(1) =1$. Or equivalently, $$\mathrm{core}(p^{2k+1})=p,\mathrm{core}(p^{2k})=1$$ Since the Dirichlet $\mathrm{core}$ generating function is: \begin{align} \begin{split} C(s)&=\sum_{n\ge 1}\frac{\mathrm{core}(n)}{n^s} =\dots=\frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)}. \end{split} \end{align}
Is it possible to use Möbius inversion How to use an inverse Mellin transform on the last equation to get $\mathrm{core}(n)$?