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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.

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  • $\begingroup$ Somehow yes, since Gaussian primes $a+bi$ are somehow uniformly distributed on the plane. $\endgroup$
    – Fedor Petrov
    Commented Sep 1, 2017 at 21:23
  • $\begingroup$ @FedorPetrov Are they? $\endgroup$
    – Igor Rivin
    Commented Sep 1, 2017 at 21:26
  • $\begingroup$ @MacNamara please look at this paper: arxiv.org/pdf/1705.07498.pdf $\endgroup$
    – Khadija Mbarki
    Commented Sep 1, 2017 at 21:38

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Here is a good paper that can answer your question! https://arxiv.org/pdf/1705.07498.pdf

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Following Fedor Petrov's comment: the arguments of Gaussian primes are known to be uniformly distributed, see Example 7.20 in Number theory: an introduction to class field theory by Kato et al (2000).

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