The polynomial X^2+Y^4 captures its primes
数论
2009-09-25 v1
摘要
This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} \Lambda(a^2 + b^4) = 4\pi^{-1}\kappa x^{3/4} (1 + O(\log\log x / \log x)) where a, b run over positive integers and \kappa = \int^1_0 (1 - t^4)^{1/2} dt = \Gamma(1/4)^2 /6\sqrt{2\pi}. Here of course, \Lambda denotes the von Mangoldt function and \Gamma the Euler gamma function.
引用
@article{arxiv.math/9811185,
title = {The polynomial X^2+Y^4 captures its primes},
author = {John Friedlander and Henryk Iwaniec},
journal= {arXiv preprint arXiv:math/9811185},
year = {2009}
}
备注
96 pages, published version, abstract added in migration