Is the average order of a product of arithmetic functions the product of the average orders?

Question

If we have $$\sum_{n \leq x} f(n) \sim \sum_{n \leq x} g(n)$$ $$\sum_{n \leq x} h(n) \sim \sum_{n \leq x} k(n)$$ does it follow that $$\sum_{n \leq x} f(n)h(n) \sim \sum_{n \leq x} g(n)k(n)$$

In other words, if $g$ is an average order of $f$ and $k$ is an average order of $h$, then is $gk$ an average order of $fh$?

Answer 1

No, for example pick $f(n) = 1+(-1)^n, h(n) = 1-(-1)^n, g(n)=k(n)=1$