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Here is a really basic question which I wished I understood better about the primes of the Gaussian field $\mathbb{Z}[i]$. But I was curious about the possibility of generalizing it to other (real quadratic or higher degree) number fields, see below.

Consider the primes $p < X$ which split $\mathbb{Q}(i)$, i.e., the ones $p \equiv 1 \mod{4}$. Then $p = a^2+b^2$ in an essentially unique way: one can switch the signs of $a$ or $b$ or interchange them, giving a total of $8$ points $(a,b)$ on the circle $|z| = \sqrt{p}$ in the Gaussian plane. This corresponds to looking at the two primes $(a+ib)$ and $(a-ib)$ above $p$ in $\mathbb{Z}[i]$, and noting that the unit group is just $\{\pm 1, \pm i\}$. Mark all these $8$ points (so that there is no ambiguity), for all the split primes $p < X$. As $X$ grows, how do these marked points distribute in angular sectors?

This sounds like Sato-Tate involving a curve $y^2 = x^3-x$ with CM by our Gaussian ring, particularly since $a \pm i b$ is a Weil number of weight one. But so also is its reflection $-i(a \pm ib) = b \mp ia$, and the two are not conjugate, hence would correspond to different elliptic isogeny classes / $\mathbb{F}_p$ in Honda's construction. (And I am confused.) Question: What is the relation, and how to solve in the most direct way possible the elementary problem here about sums of two squares?

And the second part of my question, which was also my motivation, concern the possibility of extending this to general number fields $K/\mathbb{Q}$. The prime ideal theorem describes the distribution of the norms of primes of $K$. But these prime ideals sit as lattices in the real vector space $K \otimes_{\mathbb{Q}} \mathbb{R}$, with points representing specific algebraic numbers (elements) from $K$. One could ask about the distribution of other qualities of these lattices than just their covolume (which gives essentially the norm $N (\mathfrak{p})$, as their index in the integer lattice $O_K$). As before, for each prime with $N(\mathfrak{p}) < X$, we could mark the first $\deg{K} = \dim(K \otimes_{\mathbb{Q}} \mathbb{R})$ shortest vectors in the corresponding lattice (if there are repetitions, mark them all), and ask how the marked points distribute as $X \to \infty$. The primes of degree $> 1$ make a negligible contribution, and so for $\mathbb{Q}(i)$ the question reduces to the above elementary problem.

Has anything been written on this type of question (the distribution of lattices of prime ideals, rather than just their covolumes), or is this question uninteresting for some reason that I do not see?

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    $\begingroup$ I have to run now, but $L(1,\chi)\neq 1$ for unramified Hecke characters $\chi$ of $\mathbb{Q}(i)$ implies that $(a,b)$ is equidistributed in angle (in the sense that if you take a large disc etc.) $\endgroup$ – GH from MO Sep 22 '14 at 5:49
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    $\begingroup$ The first question has been discussed here - mathoverflow.net/questions/133410/hecke-equidistribution $\endgroup$ – Asaf Sep 22 '14 at 5:55
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    $\begingroup$ Sorry, I had a typo, I meant $L(1,\chi)\neq 0$. And yes, I meant equidistribution with respect to the uniform measure. $\endgroup$ – GH from MO Sep 22 '14 at 10:27
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    $\begingroup$ The equidistribution theorem for Gaussian primes in the first quadrant is mentioned as an example in Lang's Algebraic Number Theory. See Example 2 in Chapter XV. $\endgroup$ – KConrad Sep 23 '14 at 22:26
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    $\begingroup$ A related question would be the limiting distribution of the points $(x/\sqrt{n},y/\sqrt{n})$ for solutions to $x^2+y^2=n$, but now don't require that $n$ be prime. A much harder question is the analogue for $x^3+y^3+z^3=n$ with the points now placed on the unit sphere. I just saw a wonderful lecture on this topic by Zeev Rudnick at a conference at CRM in Montreal. And he's kindly allowed his slides to be posted: crm.umontreal.ca/2014/Stats14/pdf/Rudnick_diapos3.pdf $\endgroup$ – Joe Silverman Sep 23 '14 at 22:33
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Try chapter 12 in the great book "Prime detecting sieves" by a frequent commentator in mathoverflow, Glyn Harman.

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  • $\begingroup$ Thank you! I will get my hands on this book, which seems terrific, though to my shame I had not previously heard about it. $\endgroup$ – Vesselin Dimitrov Sep 23 '14 at 20:06

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