Hyperbolic prime number theorem

John B. Friedlander Department of Mathematics, University of Toronto Henryk Iwaniec Department of Mathematics, Rutgers University

TBD mathscidoc: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$x$.
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  title={Hyperbolic prime number theorem},
  author={John B. Friedlander, and Henryk Iwaniec},
  booktitle={Acta Mathematica},
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.
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