Definition:Dirichlet Density

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Let $\PP$ be a set of prime numbers.

For $s \in \C$, let $\ds \map f s = \sum_{p \mathop \in \PP}: p^{-s}$.

$S$ has Dirichlet density $\alpha$ if and only if:

$\ds \lim_{s \mathop \to 1^+} \set {\frac {\map f s} {\map \ln {s - 1} } } = -\alpha$

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

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.