Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The **prime ideal counting function** of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm Spec}(D):N({\frak P})\le x\}.$

- For $D=\Bbb Z$, the prime ideals are the principal ideals generated by the positive prime elements, or colloquially just the "primes." The norm is trivial; $N(p\Bbb Z)=p$. The prime number theorem states the asymptotic relation $\pi_{\Bbb Z}(x)\sim\frac{x}{\log_{\Large e}x}$.
- More generally if $D={\frak O}_K$ is the ring of integers of a number field $K$, the same asymptotic relation holds for $\pi_{{\frak O}_K}$. This is known as Landau's prime ideal theorem.
- Let $D={\Bbb F}_q[T]$ where $q$ is a prime power. The explicit formula $\frac{1}{n}\sum_{d\mid n}\mu(\frac{n}{d})q^d$ for the number of monic irreducibles of degree $n$ yields the asymptotic $\pi_{{\Bbb F}_q[T]}(x)\sim\frac{x}{\log_qx}$.

Call an asymptotic relation of the form $\pi_D(x)\sim\frac{x}{\log_{\large b}x}$ a "prime ideal theorem." Some questions:

What general kinds of domains have prime ideal theorems? Any examples strikingly different from above? Is there a conceptual (say, geometric or probabilistic) explanation for why (or heuristic prediction for) the growth class looking like $\frac{x}{\log_bx}$ in (any) of these cases? What kinds of parameters $b$ are possible? Any cases different from $b\in\{e,2,3,5,\cdots\}$? What exactly would $b$ be "saying" about $D$ anyway; what does it measure or describe?