Definition:Dirichlet Density

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

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

We say that $S$ has Dirichlet density $\alpha$ if


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

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