On a theorem of Friedlander and Iwaniec

In 131, Friedlander and Iwaniec (2009) studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements [GRAPHICS] such that the norm squared parallel to gamma parallel to(2)=a(2)+b(2)+c(2)+d(2)=p, is 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 epsilon SL(2. Z[i]) such that parallel to y parallel to(2) = n. In particular, every prime is represented. The proof is an application of Siegel's mass formula. (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.