angular distribution of primes in a number field

Question

For my master's thesis, I need to work on a problem related to the "angular" distribution of primes in a Number field. For the sake of simplicity, let's take a real quadratic number field K. We have the Minkowski embedding $\varphi: K \to \mathbb{R}^2$. Regard $K$ as a subset of $\mathbb{R}^2$ through this embedding. Let's take two angles, $\theta_1$ and $\theta_2$ in the first quadrant in $\mathbb{R}^2$. I wanted to understand the relationship between the distribution of primes in two cones of angles $\theta_1$ and $\theta_2$ centered at the origin. I found this paper (https://link.springer.com/content/pdf/10.1007/BF01472215.pdf) by Hans Rademacher; however, I could not get hold of an English translation. Is anyone aware of any other paper that might deal with this question or a possible source to find a translation? Or can someone help me understand just the main result of the paper?

Answer 1

View $K$ as a subfield of $\Bbb{R}$, let $u>1$ be the fundamental unit and for $n\in \Bbb{Z},a\in K^\times$ let $$f_n(a) = \exp(2i\pi n\frac{\log|a|}{\log |u|})$$ $f_n$ is a character $K^\times \to \Bbb{C}^\times$ which is trivial on $O_K^\times$ so it extends to a character $I(K) \to \Bbb{C}^\times$:

For a fractional principal ideal let $f_n(\frac{(a)}b)= f_n(a)/f_n(b)$. Then take a non-principal ideal $J$, let $r$ be the least integer such that it is principal, set $f_n(J)$ as any $r$-th root of $f_n(J^r)$. For any ideal in the class of $J^m$ for some $m$ let $f_n(a J^m)= f_n(a) f_n(J)^m$. So $f_n$ is defined on the group of ideals $H$ generated by $J$ and the principal ideals. Take an ideal $J'$ not in $H$, take the least $r$ such that $(J')^r\in H$, let $f_n(J')$ be a $r$-th root of $f_n((J')^r)$, and so on until that $f_n$ is defined on all the ideals.

$f_n$ extends to a Hecke character, a continuous character $\Bbb{A}_K^\times/K^\times \to \Bbb{C}^\times$: for $a\in \Bbb{A}_K^\times$ set $$f_n(a) = f_n(a_{\infty,1}^{-1}) \prod_P f_n(P)^{v_P(a)}$$

The PNT for Hecke L-functions gives that $$\sum_{P\text{ prime }\subset O_K,N(P)\le x} f_n(P) = 1_{n=0} \frac{x}{\log x}+ o(\frac{x}{\log x})$$ from which, for any $b$ real $$\sum_{P\text{ prime }\subset O_K,N(P)\le x} \Re(f_1(P)e^{-ib})^n = \frac{x}{\log x}\int_0^1 \cos^n(2\pi t)dt+o(\frac{x}{\log x})$$ which implies the equidistribution in $\{ z\in \Bbb{C},|z|=1\}$ of the $f_1(P)$ (with $P$ in the prime ideals ordered by ideal norm).

Multiplying $f_1$ with the classgroup characters (which is equivalent to choosing different $r$-th roots in the extension of $f_1$ to the non-principal ideals) you'll get the same equidistribution result restricted to the principal prime ideals.