A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale.

I would like ask about the much weaker statement forgetting questions about angular bias.

**Do all sectors $(\theta, \phi)$ have infinitely many primes $ \mathfrak{p} $?**

The idea being to find cases where it is possible to mimick Euclid's proof of the infinitude of primes in $\mathbb{Z}$. I was not able to find such an elementary proof. Certainly let $|\theta - \phi| \ll \frac{\pi}{8}$.

This may be like the cause of Dirichlet's theorem of primes in arithmetic progressions, Euclid's proof extends to some arithmetic progressions (like $4k+3,6k+5, 8n+5$) and not others [1]