TY - JOUR
T1 - Sur un théorème de Friedlander et Iwaniec
AU - Bourgain, Jean
AU - Kontorovich, Alex
N1 - Funding Information:
E-mail addresses: [email protected] (J. Bourgain), [email protected] (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 -