# How many non principal prime ideals does a number field contain?

Let $$K$$ be a number field with ring of integers $$O_K$$ is not PID. Can we estimate the cardinality of the following sets $$\mathcal{A}= \{\mathcal{P}\subset O_K \ |\ Nm(\mathcal{P})\leq x, \mathcal{P}\ \text{ is not principal}\},$$ $$\mathcal{B}= \{\mathcal{P}\subset O_K \ |\ Nm(\mathcal{P})\leq x, \mathcal{P}\ \text{ is principal}\}.$$ Any hint or comments are welcome. Thanks.

• The title says "non prime" but body asks for principal and nonprincipal ideals. You probably want to edit the title. Is $\mathcal P$ supposed to be prime? It is not explicitly said. Commented Oct 29, 2020 at 11:03
• If you like my answer, please accept it officially (so that it turns green). Thanks in advance! Commented Oct 29, 2020 at 12:56

I assume that, in your question, $$\mathcal{P}$$ means a prime ideal of $$\mathcal{O}_K$$.
It follows from the non-vanishing of Hecke $$L$$-functions $$L(s,\chi)$$ at $$s=1$$ (where $$\chi$$ is an unramified Hecke character of $$K$$) that $$\#\mathcal{B}$$ is asymptotically $$\mathrm{li}(x)/h(K)$$, and $$\#\mathcal{A}$$ is asymptotically $$(1-1/h(K))\mathrm{li(x)}$$. More generally, the prime ideals of $$\mathcal{O}_K$$ are equidistributed in the ideal class group of $$K$$. The proof is the same as the proof of Dirichlet's theorem on arithmetic progressions. (Here of course $$\mathrm{li}(x)$$ can be replaced by $$x/\ln(x)$$, but $$\mathrm{li}(x)$$ is a better approximation.)
• It may be useful to add that this is a special case of the Chebotarev density theorem, applied in the case of the Hilbert class field of $K$. Commented Oct 29, 2020 at 8:48