The classical prime number theorem states that the prime counting function $$\pi(X) := \# \{ p \leq X \ | \ \text{$p$ prime} \}$$ is asymptotically equal to $X/\log(X)$.

It is also known (and much simpler to prove) that there is a finite function field analogue $$\#\{ f \in \mathbb{F}_q[T] \ | \ \text{$f$ monic, irreducible}, \ N(f) \leq X\} \sim X/\log_q(X)$$ where we consider the norm $N(f) := \#(\mathbb{f}_q[T]/(f)) = q^{\deg(f)}$.

Furthermore, we also have the Landau prime ideal theorem, which states that in a number field $K$, have $$\#\{ \mathfrak{p} \subseteq \mathcal{O}_K \ | \ \text{$\mathfrak{p}$ prime ideal}, \ N(\mathfrak{p}) \leq X \} \sim X/\log(X)$$

The similarity of these expressions is striking, and since $\mathcal{O}_K$ and $\mathbb{F}_q[T]$ (and finite extensions thereof) are the "classical" examples for Dedekind domains, it seems natural to conjecture an analogue for all/many Dedekind domains $R$.

So assuming we have a Dedekind domain $R$ such that $R/\mathfrak{p}$ is finite for all primes $\mathfrak{p}$, we might have $$\#\{ \mathfrak{p} \subseteq R \ | \ \text{$\mathfrak{p}$ prime ideal}, \ \#(R/\mathfrak{p}) \leq X \} \sim c_R X / \log(X)$$ for some constant $c_R$.

My basic question is now: Is this true/proven/conjectured?

To go futher, I would also be very interested in how this looks for more general rings? In particular, is there a class of integral rings $R$ (with $R/\mathfrak{p}$ finite for all primes $\mathfrak{p}$) that have "more" small prime ideals than the classical rings above, if we again measure the size of prime ideals via $\#(R/\mathfrak{p})$?

I have found this question, but while starting with the PNT, it is more interested in general connections of number fields and function fields.

**Edit** As Wojowu pointed out, it is clear that we can arbitrarily decrease the number of small primes by localization. Nevertheless, I am still very interested in my additional question: Are there rings that have asymptotically more than $X/\log(X)$ primes $\mathfrak{p}$ with $\#(R/\mathfrak{p}) \leq X$?