Definition:Formal Derivative of Dirichlet Series

Definition
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:
 * $\ds \sum_{n \mathop = 1}^\infty \dfrac {a_n \log n} {n^s}$

Also see

 * Derivative of Dirichlet Series
 * Definition:Formal Product of Dirichlet Series