Let $f$ and $g$ are multiplicative and $a$ and $b$ are positive integers with $a\ge b$. We can prove that $h(n)=\sum\limits_{d^a|n}f(\frac n{d^a})g(\frac n{d^b})$ is multiplicative as follows. Let $(m,n)=1$. Then we have
$$h(m)h(n)=\sum_{c^a|m}f(\frac m{c^a})g(\frac m{c^b})\sum_{d^a|n}f(\frac n{d^a})g(\frac n{d^b})\\ =\sum_{c^a|m\\d^a|n}f(\frac m{c^a})g(\frac m{c^b})f(\frac n{d^a})g(\frac n{d^b})\\ =\sum_{t^a|mn}f(\frac{mn}{t^a})g(\frac{mn}{t^b})=h(mn).$$
Thus, It seems that we didn't use $a\ge b$. But Apostol's number theory book has stated this problem for $a\ge b$. So I want to know if $a\ge b$ is necessary? Thanks!
This ensures that $d^b$ divides $n$ in the definition of $h$, so that $g$ is evaluated only where it is defined, i.e. at integers.