For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I am looking for a reference for this nice result. I would be thankful if one can give me a reference. 



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.]


