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2009 Hyperbolic prime number theorem
John B. Friedlander, Henryk Iwaniec
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Acta Math. 202(1): 1-19 (2009). DOI: 10.1007/s11511-009-0033-z


We count the number S(x) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $is a prime number and satisfying the determinant condition: x1x4 − x2x3 = 1. By means of the sieve, one shows easily the upper bound S(x) ≪ x/log x. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is S(x) ≫ x/log x.

Funding Statement

The first author is supported in part by NSERC grant A5123. The second author is supported in part by NSF grant DMS-03-01168.


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John B. Friedlander. Henryk Iwaniec. "Hyperbolic prime number theorem." Acta Math. 202 (1) 1 - 19, 2009.


Received: 12 December 2006; Published: 2009
First available in Project Euclid: 31 January 2017

zbMATH: 1278.11089
MathSciNet: MR2486486
Digital Object Identifier: 10.1007/s11511-009-0033-z

Rights: 2009 © Institut Mittag-Leffler


Vol.202 • No. 1 • 2009
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