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.



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


