Definition:Dirichlet Density

Definition
Let $\mathcal P$ be a set of prime numbers.

For $s \in \C$, let $\displaystyle f \left({s}\right) = \sum_{p \mathop \in \mathcal P}\: p^{-s}$.

$S$ has Dirichlet density $\alpha$ iff:


 * $\displaystyle \lim_{s \mathop \to 1^+} \left\{{f \left({s}\right) \ln \left({s - 1}\right)^{-1}}\right\} = - \alpha$

where $1^+$ indicates a limit from above along the real line.