I have been wondering about the distribution in the complex plane of Gaussian primes -- essentially those numbers $a + i b$ where $a, b$ are integers and $a^2+b^2$ is a prime number [I know there is another sort but confined on the real line]. Examples are $5+2i$ or $4+i$ (signums are irrelevant of course).
My initial query was about the distribution of phases, i.e. arguments (say $\arctan b/a$), which seems to be uniform -- though one would need some clearcut definition of ``uniform'' if one wants to check, say, Weyl's criterion here.
Another reasonable conjecture is that there exists such Gaussian primes for any given $a$ (resp. $b$). Perhaps even infinitely many? This latter conjectures includes existing conjectures, like whether there exists infinitely many primes succeeding squares, e.g. $p= a^2+1$.
I have looked around and found a question here but it does really answer mine. The last conjecture appears also here and is a special case of Schintzel's H hypothesis.
This distribution has some inherent beauty if nothing else:
