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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?

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The term you want to look up is "Abstract Analytic Number Theory". John Knopfmacher wrote a book with this title. See also en.wikipedia.org/wiki/Abstract_analytic_number_theory –  KConrad Apr 9 at 3:48
    
Also search for "zeta functions of groups and rings". –  Daniel Loughran Apr 9 at 8:18
    
For your function $\pi_D(x)$ to be defined, you need every non-zero prime ideal $P$ to be such that $D/P$ is finite. Since a finite domain is a field, that means that the ting has to be of dimension 1. Therefore your example pretty much cover the whole spectrum of possibility, and I don't such another behavior for $\pi_D(x)$ should be expected. Now, perhaps you would be interested in generalizing your question as follows: take $D$ any domain which is of finite type over $\mathbb Z$, and let $\pi_D(x)$ be the number of maximal ideals $M$ such that $|D/M|<x$ ($D/M$ is finite, Nullstellensatz) –  Joël Apr 9 at 13:38
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Joel, I think there is no problem with the definition if there is a $P$ such that $D/P$ is infinite. The primes of infinite norm will just never be counted. –  Maarten Derickx Apr 9 at 21:06

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