On the Probability of Relative Primality in the Gaussian Integers
arXiv:1305.5502
Abstract
This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{ζ_{\Q(i)}(2)}$$ where are randomly chosen and $ζ_{\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theorem to arbitrary number fields that are principal ideal domains.
6 pages, 1 figure