Hyperbolic prime number theorem

Abstract

We count the number S(x) of quadruples \left( x-1 ,x-2 ,x-3 ,x-4 \right) \in \mathbbZ4 for which p = x2-1 + x2-2 + x2-3 + x2-4 \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. © 2009 Institut Mittag-Leffler.