Natural and Dirichlet density

Let $A\subseteq {\mathbb{N}}_{\ge 1}$ and write $A(x):=\mathrm{\#}\{n\le x:n\in A\}$.

• If the limit exists, the natural density of $A$ is ${\lim }_{x\to \mathrm{\infty }}\frac{A(x)}{x}\mathrm{.}$
• If the limit exists, the Dirichlet density of $A$ is the number $\delta$ for which ${\lim }_{s\downarrow 1,s>1}(s-1){\sum }_{n\in A}{n}^{-s}=\delta ,$ equivalently ${\sum }_{n\in A}{n}^{-s}=\frac{\delta }{s-1}+o\text{\hspace{0.17em}⁣}\left(\frac{1}{s-1}\right)(s\downarrow 1,s>1)\mathrm{.}$

If $P$ is a set of primes, its relative natural density among the primes is the limit of ${\pi }_P(x)\mathrm{/}\pi (x)$ when that limit exists, where ${\pi }_P(x):=\mathrm{\#}\{p\le x:p\in P\}$.

Its relative Dirichlet density among the primes is the number $\delta$ for which

${\lim }_{s\downarrow 1,s>1}\frac{{\sum }_{p\in P}{p}^{-s}}{{\sum }_p{p}^{-s}}=\delta \mathrm{.}$

Since ${\sum }_p{p}^{-s}=\log (1\mathrm{/}(s-1))+O(1)$ as $s\downarrow 1$, this is equivalently

${\sum }_{p\in P}{p}^{-s}=\delta \log \frac{1}{s-1}+o\text{\hspace{0.17em}⁣}(\log \frac{1}{s-1})(s\downarrow 1,s>1)\mathrm{.}$