  In [FI09], Friedlander and Iwaniec studied the so-called Hyperbolic Prime
Number Theorem, which asks for an infinitude of elements gamma in SL(2,Z) such
that the norm squared |gamma|^2 = a^2 + b^2 + c^2 + d^2 = p, a prime. Under the
Elliott-Halberstam conjecture, they proved the existence of such, as well as a
formula for their count, off by a constant from the conjectured asymptotic. In
this note, we study the analogous question replacing the integers with the
Gaussian integers. We prove unconditionally that for every odd n>=3, there is a
gamma in SL(2,Z[i]) such that |gamma|^2 = n. In particular, every prime is
represented. The proof is an application of Siegel's mass formula.
