Number Theory
On a theorem of Friedlander and Iwaniec
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 947-950.

In [3], Friedlander and Iwaniec (2009) studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements $γ=(abcd)∈SL(2,Z)$ such that the norm squared

 $‖γ‖2=a2+b2+c2+d2=p,$
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 $γ∈SL(2,Z[i])$ such that $‖γ‖2=n$. In particular, every prime is represented. The proof is an application of Siegel's mass formula.

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

 $‖γ‖2=a2+b2+c2+d2=p,$
$γ=(abcd)∈SL(2,Z)$. 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 $γ∈SL(2,Z[i])$, et montrons que $‖γ‖2$ représente alors en fait tout nombre impair. La formule de masse de Siegel joue un rôle essentiel.

Published online:
DOI: 10.1016/j.crma.2010.08.004
Jean Bourgain 1; Alex Kontorovich 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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Jean Bourgain; Alex Kontorovich. On a theorem of Friedlander and Iwaniec. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 947-950. doi : 10.1016/j.crma.2010.08.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.08.004/

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

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

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

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

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

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

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

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

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

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