# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332000

Acta Mathematica, 202, (1), 1-19, 2006.12
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$x$.
@inproceedings{john2006hyperbolic,
title={Hyperbolic prime number theorem},
author={John B. Friedlander, and Henryk Iwaniec},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203352727616709},
booktitle={Acta Mathematica},
volume={202},
number={1},
pages={1-19},
year={2006},
}

John B. Friedlander, and Henryk Iwaniec. Hyperbolic prime number theorem. 2006. Vol. 202. In Acta Mathematica. pp.1-19. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203352727616709.