Hyperbolic prime number theorem

We count the number S ( x ) of quadruples for which 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–Vinograd...

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Bibliographic Details
Published in:Acta mathematica 2009-03, Vol.202 (1), p.1-19
Main Authors: Friedlander, John B., Iwaniec, Henryk
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
General mathematics
General, history and biography
Mathematics
Mathematics and Statistics
Number theory
Prime numbers
Sciences and techniques of general use
Studies
Theorems
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Summary:We count the number S ( x ) of quadruples for which 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 .
ISSN:0001-5962
1871-2509
DOI:10.1007/s11511-009-0033-z