On a theorem of Friedlander and Iwaniec

Bourgain, Jean; Kontorovich, Alex

doi:10.1016/j.crma.2010.08.004

Number Theory

On a theorem of Friedlander and Iwaniec
[Sur un théorème de Friedlander et Iwaniec]

Présenté par : Christophe Soulé

Bourgain, Jean ¹ ; Kontorovich, Alex ^1,²

¹ Department of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
² Department of Mathematics, Brown University, Providence, RI 02912, USA

Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 947-950.

Résumé
Abstract

Dans [3], Friedlander et Iwaniec (2009) ont introduit l'ensemble des nombres premiers qui admettent une représentation

‖ γ ‖^{2} = a^{2} + b^{2} + c^{2} + d^{2} = p,

où

γ = (\begin{matrix} a & b \\ c & d \end{matrix}) \in SL (2, Z)

. Ils y étudient la question de savoir si cet ensemble est infini, et le démontrent sous la conjecture de Elliott et Halberstam. Dans cette Note, nous considérons le problème analogue pour les entiers de Gauss, donc

γ \in SL (2, Z [i])

, et montrons que

‖ γ ‖^{2}

représente alors en fait tout nombre impair. La formule de masse de Siegel joue un rôle essentiel.

In [3], Friedlander and Iwaniec (2009) studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements $γ = (\begin{matrix} a & b \\ c & d \end{matrix}) \in SL (2, Z)$ 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 \in 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.

Reçu le : 2010-08-05
Publié le : 2010-09-01

DOI : 10.1016/j.crma.2010.08.004

Affiliations des auteurs :

Bourgain, Jean ¹ ; Kontorovich, Alex ^{1, 2}

¹ Department of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
² Department of Mathematics, Brown University, Providence, RI 02912, USA

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     title = {On a theorem of {Friedlander} and {Iwaniec}},
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Bourgain, Jean; Kontorovich, Alex. On a theorem of Friedlander and Iwaniec. Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 947-950. doi : 10.1016/j.crma.2010.08.004. http://www.numdam.org/articles/10.1016/j.crma.2010.08.004/

Bibliographie
Cité par

[1] Bourgain, Jean; Gamburd, Alex; Sarnak, Peter Sieving and expanders, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006) no. 3, pp. 155-159

[2] Cassels, J.W.S. Rational Quadratic Forms, London Mathematical Society Monographs, vol. 13, Academic Press, London–New York–San Francisco, 1978

[3] Friedlander, John B.; Iwaniec, Henryk Hyperbolic prime number theorem, Acta Math., Volume 202 (2009) no. 1, pp. 1-19

[4] Iwaniec, Henryk Almost-primes represented by quadratic polynomials, Invent. Math., Volume 47 (1978), pp. 171-188

[5] Kitaoka, Yoshiyuki A note on local densities of quadratic forms, Nagoya Math. J., Volume 92 (1983), pp. 145-152

[6] Reiner, Irma On the two-adic density of representations by quadratic forms, Pacific J. Math., Volume 6 (1956), pp. 753-762

[7] Siegel, Carl Ludwig Über die analytische Theorie der quadratischen Formen, Ann. Math. (2), Volume 36 (1935) no. 3, pp. 527-606

[8] Yang, Tonghai An explicit formula for local densities of quadratic forms, J. Number Theory, Volume 72 (1998) no. 2, pp. 309-356

[9] Yang, Tonghai Local densities of 2-adic quadratic forms, J. Number Theory, Volume 108 (2004) no. 2, pp. 287-345

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