Dirichlet density

Let be an algebraic number field (cf. also Algebraic number) and let be a set of prime ideals (of the ring of integers ) of . If an equality of the form

holds, where is regular in the closed half-plane , then is a regular set of prime ideals and is called its Dirichlet density. Here, is the norm of , i.e. the number of elements of the residue field .

Examples.

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

ii) Let be a finite extension and the set of all prime ideals in that are of degree over (i.e. , where is the prime ideal under ). Then is regular with Dirichlet density .

iii) Let be a finite normal extension and the set of all prime ideals in that split in (i.e. is a product of prime ideals in of degree ). Then is regular with Dirichlet density .

The notion of a Dirichlet density can be extended to not necessarily regular sets of prime ideals. Such a set has Dirichlet density if

References

[a1]	W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.2 (Edition: Second)