Hyperbolic prime number theorem

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: 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.