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.