On a Theorem of Friedlander and Iwaniec

TL;DR

This paper proves that for every odd integer greater than or equal to 3, there exists a matrix in SL(2,Z[i]) with a norm squared equal to that integer, extending prime representation results to Gaussian integers.

Contribution

It establishes an unconditional result that every odd integer ≥ 3 can be represented as the norm squared of a matrix in SL(2,Z[i]), generalizing previous prime-related theorems.

Findings

Every odd n ≥ 3 is represented by some gamma in SL(2,Z[i])

In particular, all primes are represented

Uses Siegel's mass formula for the proof

Abstract

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.