Definition:Dirichlet Density

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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 $\alpha$ if and only if:

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

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

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.