Asymptotic for $\sum a_nb_n$ if asymptotic for $\sum a_n, \sum b_n$ are known

Question

Let us assume that $a_n>0$ and $b_n>0$ for each n. Also let $$ \sum_{n\leq x} a_n \sim f(x) $$ and $$ \sum_{n\leq x} b_n \sim g(x) $$. What can we say about the asymptotic on $\sum_{n \leq x} a_nb_n$?

Answer 1

Without knowing more of the structure of the sequences $a_n$ and $b_n$, you cannot say anything about the asymptotic for $\sum_{n\leq x} a_n b_n$, except the simple upper bound $\sum_{n\leq x} a_n b_n\leq f(x)g(x)$. In fact, without any restrictions on $a_n, b_n$ if $f,g$ are monotonic, then for any epsilon, the asymptotic function $h(x)$ for $\sum_{n\leq x} a_n b_n\leq f(x)g(x)$ can be any monotonic function satisfying $0< h(x)\leq (f(x)g(x))^{1-\epsilon}$.