I would like to know the asymptotics for $n \to \infty$ of the two functions $$\sum_{d \mid n} \frac{\mu(d)}{d^2}n^2$$
and $$\sum_{d \mid n} \frac{\mu^2(d)}{d} n.$$
I see nothing else to do than to write the expanded sum and... this is not enough to "see" the main term. Is there any general method to estimate asymptotics for such convolutions?
Those things don't have a simple asymptotic because
$$\sum_{d |n} \frac{\mu(d)^r}{ d^2}n =n\prod_{p^k \| n} \sum_{d |p^k}\frac{\mu(d)^r}{ d^2}=n\prod_{p | n} (1+ \frac{(-1)^r}{p^2})$$ With $n=m!$ and $m$ large then $\prod_{p | n} (1+ \frac{(-1)^r}{p^2})$ will be close to $1/\zeta(2)$ if $r$ is odd or $\zeta(2)/\zeta(4)$ if $r$ is even, whereas for $n$ prime $\prod_{p | n} (1+ \frac{(-1)^r}{p^2})$ will be close to $1$.