TY - JOUR

T1 - Sur un théorème de Friedlander et Iwaniec

AU - Bourgain, Jean

AU - Kontorovich, Alex

N1 - Funding Information:
E-mail addresses: bourgain@ias.edu (J. Bourgain), avk@math.ias.edu (A. Kontorovich). 1 Partially supported by NSF grant DMS-0808042. 2 Partially supported by NSF grants DMS-0802998 and DMS-0635607, and the Ellentuck Fund at IAS.

PY - 2010/9

Y1 - 2010/9

N2 - 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.

AB - 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.

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U2 - 10.1016/j.crma.2010.08.004

DO - 10.1016/j.crma.2010.08.004

M3 - Article

AN - SCOPUS:77957156537

SN - 1631-073X

VL - 348

SP - 947

EP - 950

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

IS - 17-18

ER -