Hecke equidistribution

One reference is Theorem 6 of Chapter XV (Density of Primes and Tauberian Theorem) in

S. Lang: Algebraic Number Theory (Addison-Wesley, 1970).

This is probably more general than Hecke's result, but the case of "equidistribution of ideals and primes in sectors" of the Gaussian numbers is singled out as Example 2 on page 318.

[No, I didn't know this off the top of my head; my student David Jao needed this result in the case of a real quadratic field for his thesis in 2003, and I looked in the bibliography to find that he used the Lang reference $-$ or more accurately its second edition (1994) by Springer.]

A very down-to-earth treatment of this result of Hecke is in Chapter 5 of the nice book Geometric and Analytic Number Theory, by Hlawka, Schoißengeier, and Taschner. By down-to-earth, I mean that they deal directly with this specific case of Hecke's result, and that they prove it using very little -- the method is a modification of the Korevaar--Newman--Zagier approach to the prime number theorem, and so doesn't need any quantitative zero-free region (just a statement that there are no zeros of the appropriate objects on the line $\Re(s)=1$).

If you need the original reference, this is proved in Hecke's articles (here and here) where he introduces the famous $L$-functions associated to Grössencharakteren.

E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Math. Z. 6 (1920), no. 1-2, 11--51 ; Math. Z. 1 (1918), no. 4, 357--376.

Here is an extract from the Zentralblatt review of Hecke's articles :

Übersetzt man den so gefundenen Sachverhalt in die Sprache der Formentheorie, so ergeben sich Sätze von folgender Art: Jede der Formen $x^2+y^2$ und $x^2−2y^2$ stellt unendlich viele Primzahlen dar, wenn man die Variabeln $x$, $y$ auf eine beliebigen Winkelraum einschränkt, welcher von zwei vom Nullpunkt der $x$−$y$-Ebene ausgehenden Halbstrahlen begrenzt wird. Überdies ist die Anzahl dieser Primzahlen unterhalb $t$ für $t \to > \infty$ asymptotisch proportional der Grösse dieses Winkels, gemessen in einer auf die betreffende Form gegründeten Klein-Cayleyschen Massbestimmung.