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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 $M(a+bi) = a^2+b^2$.

Mark Fischler
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    Knopfmacher's book "Abstract Analytic Number Theory" considers this case and many others. There exists a version of Prime Number Theorem for Gaussian integers. – Ángel Valencia Mar 18 '16 at 06:00
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    Yeah, google Landau's prime number theorem – Arkady Mar 18 '16 at 06:05
  • Landau's Prime Ideal Theorem, at least in the form I found, only gives the asymptotic density along the real axis. While it is certainly plausible that the density along any other direction is the same, does knopfmacher's book give a proof of that? – Mark Fischler Mar 18 '16 at 06:12
  • I'm guessing that if for prime $p = a + bi$ you only look at $p$ with $a > b \geq 0$, then you'll have a very slightly different version of the prime number theorem. – Robert Soupe Apr 13 '16 at 02:51

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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.

Greg Kuperberg
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