On a theorem of Friedlander and Iwaniec

Abstract

In [3], Friedlander and Iwaniec (2009) studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements γ = ( a b c d ) ∈ SL ( 2 , Z ) such that the norm squared ‖ γ ‖ 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 γ ∈ 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.