On the Probability of Relative Primality in the Gaussian Integers
نویسندگان
چکیده
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(z1, z2) = 1) = 1 ζQ(i)(2) where z1, z2 ∈ Z[i] are randomly chosen and ζQ(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.
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