Definition:Formal Derivative of Dirichlet Series

Definition
Let $(a_n) : \N \to \C$ be an arithmetic function.

Let $F(s)=\displaystyle \sum_{n\mathop=1}^\infty\frac{a_n}{n^s}$ be the corresponding formal Dirichlet series.

The formal derivative of $F(s)$ is the formal Dirichlet series:
 * $\displaystyle \sum_{n\mathop=1}^\infty\frac{a_n\cdot \log n}{n^s}$

Also see

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