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
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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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