\begin{definition}[Definition:Formal Derivative of Dirichlet Series]
Let $\sequence {a_n}: \N \to \C$ be an arithmetic function.
Let $\map F s = \ds \sum_{n \mathop = 1}^\infty \dfrac {a_n} {n^s}$ be the corresponding formal Dirichlet series.
The '''formal derivative''' of $\map F s$ is the formal Dirichlet series:
:$\ds \sum_{n \mathop = 1}^\infty \dfrac {a_n \log n} {n^s}$
\end{definition}