Dirichlet density

2020 Mathematics Subject Classification: Primary: 11R44 [MSN][ZBL]

Let $K$ be an algebraic number field (cf. also Algebraic number) and let $A$ be a set of prime ideals (of the ring of integers $A_K$) of $K$. If an equality of the form $$ \sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} = a \log\frac{1}{1-s} + g(s) $$ holds, where $g(s)$ is regular in the closed half-plane $\mathrm{Re}(s) \ge 1$, then $A$ is a regular set of prime ideals and $a$ is called its Dirichlet density. Here, $N(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the number of elements of the residue field $A_k/\mathfrak{p}$.

Examples.

i) The set of all prime ideals of $K$ is regular with Dirichlet density $1$.

ii) Let $L/K$ be a finite extension and $A$ the set of all prime ideals $\mathfrak{P}$ in $L$ that are of degree $1$ over $K$ (i.e. $[A_L/\mathfrak{P} : A_K/\mathfrak{p}] = 1$, where $\mathfrak{p}$ is the prime ideal $\mathfrak{P} \cap A_K$ under $\mathfrak{P}$). Then $A$ is regular with Dirichlet density $1$.

iii) Let $L/K$ be a finite normal extension and $A$ the set of all prime ideals $\mathfrak{p}$ in $K$ that split in $L$ (i.e. $\mathfrak{p}A_L$ is a product of $[L:K]$ prime ideals in $L$ of degree $1$). Then $A$ is regular with Dirichlet density $[L:K]^{-1}$.

The notion of a Dirichlet density can be extended to not necessarily regular sets of prime ideals. Such a set $A$ has Dirichlet density $a$ if $$ \lim_{s \searrow 1} \frac{ \sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} }{ a \log\frac{1}{1-s} } \ . $$

References

[a1]	W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" (2nd ed.) PWN/Springer (1990) ISBN 3-540-51250-0 Zbl 0717.11045 Sect. 7.2