I was reading about Gaussian primes when I came across an article on bounded gaps and distance between such numbers. Since $\mathbb{C}$ is not a totally ordered set, I find it strange to talk about "prime gaps" and "distances" between Gaussian numbers. In particular:
- Since it is not possible to define an operator $<$ (or $>$) such that $\forall a,b\in\mathbb{C}: a<b$, how is possible to talk about a Gaussian prime being "smaller" or "larger" than another one, or about a "distance" between consecutive Gaussian primes? Is this somehow related to geometric distances between lattice points in $\mathbb{C}$?
- What is the current research on "distances between Gaussian primes" and is it related in some manner to gaps between primes in $\mathbb{N}$?