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Hyperbolic prime number theorem

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Acta Mathematica

Abstract

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: x 1 x 4 − x 2 x 3 = 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    John B. Friedlander

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08903, USA

    Henryk Iwaniec

Authors
  1. John B. Friedlander
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  2. Henryk Iwaniec
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Corresponding author

Correspondence to John B. Friedlander.

Additional information

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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Friedlander, J.B., Iwaniec, H. Hyperbolic prime number theorem. Acta Math 202, 1–19 (2009). https://doi.org/10.1007/s11511-009-0033-z

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  • DOI: https://doi.org/10.1007/s11511-009-0033-z

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