  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 $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{\zeta_{\Q(i)}(2)}$$ where
$z_{1},z_{2} \in \mathbb{Z}[i]$ are randomly chosen and $\zeta_{\Q(i)}(s)$ 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.
