# A distribution related to Fermat's two squares theorem

Fermat's two squares theorem tell us that every prime number $p \equiv 1 \pmod 4$ can be written in a unique way as $p = a^2 + b^2$ for two positive integers $a < b$. In particular, we can associate to $p$ an angle $\theta_p = \arctan \frac{a}{b}$.

I am asking if it is known some result on the distribution of the values $\theta_p$ in the interval $[0, \frac{\pi}{2}]$ as $p$ runs over the primes $p \leq x$, $p \equiv 1 \bmod 4$, and as $x$ goes to infinity.

I guess the should be somehow uniformly distributed.

• Somehow yes, since Gaussian primes $a+bi$ are somehow uniformly distributed on the plane. – Fedor Petrov Sep 1 '17 at 21:23
• @FedorPetrov Are they? – Igor Rivin Sep 1 '17 at 21:26
• @MacNamara please look at this paper: arxiv.org/pdf/1705.07498.pdf – Khadija Mbarki Sep 1 '17 at 21:38