The $n$-th prime asymptotically

• [1] Universidad de Sevilla Facultad de Matemáticas Apdo. 1160, 41080-Sevilla Spain
• [2] University of Luxembourg, Campus Kirchberg Mathematics Research Unit, BLG 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg Grand Duchy of Luxembourg
• Volume: 25, Issue: 3, page 521-555
• ISSN: 1246-7405

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Abstract

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A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994).Realistic bounds for the error with ${li}^{-1}\left(n\right)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible ${r}_{3}$ such that, for $n\ge {r}_{3}$, we have ${p}_{n}>{s}_{3}\left(n\right)$ where ${s}_{3}\left(n\right)$ is the sum of the first four terms of the asymptotic expansion.

How to cite

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Arias de Reyna, Juan, and Toulisse, Jérémy. "The $n$-th prime asymptotically." Journal de Théorie des Nombres de Bordeaux 25.3 (2013): 521-555. <http://eudml.org/doc/275705>.

@article{AriasdeReyna2013,
abstract = {A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994).Realistic bounds for the error with $\operatorname\{li\}^\{-1\}(n)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n&gt; s_3(n)$ where $s_3(n)$ is the sum of the first four terms of the asymptotic expansion.},
affiliation = {Universidad de Sevilla Facultad de Matemáticas Apdo. 1160, 41080-Sevilla Spain; University of Luxembourg, Campus Kirchberg Mathematics Research Unit, BLG 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg Grand Duchy of Luxembourg},
author = {Arias de Reyna, Juan, Toulisse, Jérémy},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {-th prime; asymptotic expansion; Riemann hypothesis},
language = {eng},
month = {11},
number = {3},
pages = {521-555},
publisher = {Société Arithmétique de Bordeaux},
title = {The $n$-th prime asymptotically},
url = {http://eudml.org/doc/275705},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Arias de Reyna, Juan
AU - Toulisse, Jérémy
TI - The $n$-th prime asymptotically
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/11//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 3
SP - 521
EP - 555
AB - A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994).Realistic bounds for the error with $\operatorname{li}^{-1}(n)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n&gt; s_3(n)$ where $s_3(n)$ is the sum of the first four terms of the asymptotic expansion.
LA - eng
KW - -th prime; asymptotic expansion; Riemann hypothesis
UR - http://eudml.org/doc/275705
ER -

References

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