= Solution
The <Dirichlet convolution> is $(f_1*f_2)(n)=\sum_{d\mid n}f_1(d)f_2(n/d)$. Fix $s$ with $\sigma=\operatorname{Re}s>1$. <Absolute convergence> of the two <Dirichlet series> gives
$$
\sum_{a,b\ge1}|f_1(a)f_2(b)(ab)^{-s}|
=\left(\sum_a|f_1(a)|a^{-\sigma}\right)
\left(\sum_b|f_2(b)|b^{-\sigma}\right)<\infty.
$$
Thus the double <series> may be regrouped by $n=ab$, proving
$$
\boxed{F_1(s)F_2(s)=\sum_{n\ge1}(f_1*f_2)(n)n^{-s}.}
$$
The same estimate proves <absolute convergence> of the resulting <Dirichlet series>.
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