\begin{definition}[Definition:Formal Product of Dirichlet Series]
Let $f, g : \N \to \C$ be arithmetic functions.
Let $F, G$ be their formal Dirichlet series.
The '''formal product''' of $F$ and $G$ is the formal Dirichlet series:
:$\ds \sum_{n \mathop = 1}^\infty \frac {\map {\paren {f * g} } n} {n^s}$
where $*$ denotes Dirichlet convolution.
\end{definition}