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