On a Theorem of Friedlander and Iwaniec
Number Theory
2010-08-05 v1
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.
Keywords
Cite
@article{arxiv.1008.0825,
title = {On a Theorem of Friedlander and Iwaniec},
author = {Jean Bourgain and Alex Kontorovich},
journal= {arXiv preprint arXiv:1008.0825},
year = {2010}
}
Comments
4 pages