# A variant of the equidistribution of primes in an imaginary quadratic number ring

It is known that the arguments of prime elements of $\mathbb{Z}[i]$ are equidistributed in $(0,2π)$ (by Theorem 5.36 of Iwaniec and Kowalski, or one of Kubilius' papers cited below). This theorem extends to any imaginary quadratic number ring $\mathcal{O}$ if one uses prime ideal numbers (especially for those number rings that are not UFDs) as mentioned for instance in Dias's paper (cited below).

One way to remove the reliance on prime ideal numbers is to restrict the set of prime ideals under consideration to those arising from rational primes splitting into principal prime ideals (so that the prime ideal numbers are associates to the generators of these prime ideals; moreover the generators themselves are prime numbers in $\mathcal{O}$).

Now I can finally pose my question: Can someone refer me to a reference/proof in the literature that states that the prime elements arising from a rational prime splitting into principal ideals are also equidistributed in $(0,π/U)$ where $U$ denotes the number of units in $\mathcal{O}$? As far as I understand, this should be provable by applying Fourier analysis to the Chebotarev corollary I stated above (instead of the full Prime Ideal Theorem) to pick off the primes in a given sector $(\alpha, \beta) \subset (0, 2\pi)$. This process should yield an asymptotic formula of the form $\frac{\beta - \alpha}{2\pi} \cdot \frac{1}{2h} \frac{x}{\log{x}}$. Am I right about this?

This is the first time I am posting something of this magnitude on mathoverflow; so I hope I phrased it properly enough to convey what I am asking. I will happy fix or clarify anything that may be a bit imprecise. Thank you!

Remark: I am asking about the existence of this theorem in the literature so that I don't have to unnecessarily reprove it for an article I am writing.

References:

1) D. Dias, The angular distribution of integral ideal numbers with a fixed norm in quadratic extensions, 2014, available at http://arxiv.org/pdf/1404.6271v1.pdf.

2) J. Kubilius, The distribution of Gaussian primes in sectors and contours, Leningrad. Gos. Univ. U\v{c}. Zap, Cer. Mat. Nauk 137 (19) (1950) 40-52.

3) J. Kubilius, On some problems of the geometry of prime numbers, Mat. Sbornik N.S. 31 (73) (1952) 507-542.

• Angular equidistribution holds within every ideal class of a complex quadratic ring $\mathcal{O}$: for every fractional ideal $\mathfrak{b}$, the arguments of the elements $\alpha \in K = \mathrm{frak}(\mathcal{O})$ for which $\mathfrak{b}(\alpha)$ is a prime ideal are equidistributed in the circle. Have you looked into Number Theory 2 of the "Japanese" series? (by Kato, Kurokawa and Saito). I recall seeing the statement proved there in this (and more) generality. But I do not have a copy of the book handy at the moment. – Vesselin Dimitrov Jul 2 '16 at 18:26
• This result is due to Hecke. You will also find it in his Mathematische Werke, if you read German and have an access to them. Your question is the particular case of $\mathfrak{b} = (1)$, the unit ideal. – Vesselin Dimitrov Jul 2 '16 at 19:01
• I have located the angular equidistribution results for the full set of prime ideals in imaginary quadratic rings in Hecke. Are you saying that the Chebotarev variant I asserted above is a special case of Hecke or in Number Theory 2? The latter reference seems to state the prime number results in a fair amount of abstraction; how would my Chebotarev variant arise from this reference? – BDS Jul 2 '16 at 19:15
• The statement you give is contained by (52) on page 38 of Hecke's Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen (Zweite Mitteilung) (1920, Math. Zeitschrift). I recall this being worked out also as a special case of the more general equidistribution results exposed in Number Theory 2, but I don't have the book with me to check. – Vesselin Dimitrov Jul 2 '16 at 19:49
• We use, directly, the quadratic field and its Hecke $L$-functions. $\mathfrak{I}$ is, in our case, just an interval in the circle, as well as its length: your quantity $(\beta-\alpha)/2\pi$. Yes, we take just $\mathfrak{f} = (1)$ (unit conductor). Then $h_0(\mathfrak{f})$ is just the class number, and the additional $2$'' of your formula accounts for the difference between counting rational primes and counting prime ideals of bounded norm. (There are two prime ideals of norm $p$ for each split prime $p$, while the inert and unramified primes are negligible in the formula.) – Vesselin Dimitrov Jul 2 '16 at 23:57