TITLE: What is known about the counting function of Gaussian primes"
QUESTION [9 upvotes]: The counting function of primes among $\Bbb{N}$, describing the asymptotic density of the primes, is well known (the Prime Number theorem). Let's define a mild generalization of the counting function concept:
Given sets $S$ and $P\subset S$, and a map $M: S \mapsto \Bbb{N}$, then $C(S,P;M)$ is the asymptotic ratio of the number of $p \in P : M(p) \leq n$ to the number of $s \in S : M(s) \leq n$.  
For example, if we take $S$ to be $\Bbb{N}$ and $P$ to be the primes, and $M_0$ to be the trivial map $\forall k \in  \Bbb{N}, M_1(k)=k$, then the Prime Number Theorem supplies the form of $C(\Bbb{N}, $P$; M_1)$.  You could equally well transform that to a counting function $C(\Bbb{N}, $P$; M_2)$ where $\forall k \in  \Bbb{N}, M_2(k)=k^2$ for example; no interesting new information emerges.
Given this definition, my question is:  Has anybody studied the counting function of Gaussian primes?  Here, a natural map to use is the squared magnitude $M(a+bi) = a^2+b^2$.

REPLY [2 votes]: I believe that Hecke proved in 1919 that the von Mangoldt measure on Gaussian integers converges under rescaling to uniform measure in the complex plane.   Here the von Mangoldt function or measure assigns $\ln (a^2+b^2)$ to every Gaussian prime power $(a+ib)^n$.  This is therefore a 2-dimensional prime number theorem.  (As in the usual prime number theorem, you can discard the measure for higher prime powers with negligible loss of measure.)  It is usually stated in slightly different terms, but I believe that this is what it amounts to.   See for instance Rudnick and Waxman.