Reference : The $n$-th prime asymptotically |

Scientific journals : Article | |||

Physical, chemical, mathematical & earth Sciences : Mathematics | |||

http://hdl.handle.net/10993/18415 | |||

The $n$-th prime asymptotically | |

English | |

Arias de Reyna, Juan [] | |

Toulisse, Jérémy [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |

2013 | |

Journal de Théorie des Nombres de Bordeaux | |

Université de Bordeaux. Centre de Recherches Mathématiques | |

Yes (verified by ORBi^{lu}) | |

1246-7405 | |

Talence | |

France | |

[en] A new derivation of the classic asymptotic expansion
of the n-th prime is presented. A fast algorithm for the compu- tation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with $li−1 (n)$, after having re- tained the first $m$ terms, for $1 ≤ m ≤ 11$, are given. Finally, as- suming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n ≥ r_3$ , we have $p_n > s_3 (n)$ where $s_3 (n)$ is the sum of the first four terms of the asymptotic expansion. | |

http://hdl.handle.net/10993/18415 |

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