# Angles of Gaussian primes

## TL;DR

This paper investigates the distribution and variance of angles associated with Gaussian primes, extending classical results and proposing a conjecture supported by a function field analogue and random matrix models.

## Contribution

It introduces a conjecture for the variance of angles of Gaussian primes in short arcs, supported by a proven analogue in function fields and connections to random matrix theory.

## Key findings

- Angles are uniformly distributed as primes vary.
- A conjecture for the variance in short arcs is proposed.
- Asymptotic form of the variance is proved in the function field case.

## Abstract

Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$. To any of the 8 possible representations (a,b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950's Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance.