Sur un théorème de Friedlander et Iwaniec

In [3], Friedlander and Iwaniec (2009) studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements γ=(abcd)εSL(2,Z) such that the norm squared. ||γ||²=a²+b²+c²+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 γepsi;SL(2,Z[i]) such that ||γ||²=n. In particular, every prime is represented. The proof is an application of Siegel's mass formula.