The Gaussian zoo

We find all the maximal admissible connected sets of Gaussian primes: there are 52 of them. Our catalog corrects some errors in the literature. We als o describe a totally automated procedure to determine the heuristic estimat es for how often various patterns, in either the integers or Gaussian integ ers, occur in the primes. This heuristic requires a generalization of a cla ssical formula of Mertens to the Gaussian integers, which we derive from a formula of Uchiyama regarding an Euler product that involves only primes co ngruent to 1 (mod 4).