J. Sondow. (2009)cite arxiv:0907.5232Comment: 7 pages, cited Shapiro's book for Ramanujan's proof of Bertrand's Postulate.
Abstract
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if
$x R_n$, then there are at least $n$ primes in the interval $(x/2,x$. For
example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists
and gave the first five values as 2, 11, 17, 29, 41. In this note, we use
inequalities of Rosser and Schoenfeld to prove that $2n 2n < R_n < 4n łog
4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is
asymptotic to the $2n$th prime. We also estimate the length of the longest
string of consecutive Ramanujan primes among the first $n$ primes, explain why
there are more twin Ramanujan primes than expected, and make three conjectures
(the first has since been proved by S. Laishram).
%0 Generic
%1 sondow2009ramanujan
%A Sondow, Jonathan
%D 2009
%K postulate ramanujan mathematics
%T Ramanujan Primes and Bertrand's Postulate
%U http://arxiv.org/abs/0907.5232
%X The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if
$x R_n$, then there are at least $n$ primes in the interval $(x/2,x$. For
example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists
and gave the first five values as 2, 11, 17, 29, 41. In this note, we use
inequalities of Rosser and Schoenfeld to prove that $2n 2n < R_n < 4n łog
4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is
asymptotic to the $2n$th prime. We also estimate the length of the longest
string of consecutive Ramanujan primes among the first $n$ primes, explain why
there are more twin Ramanujan primes than expected, and make three conjectures
(the first has since been proved by S. Laishram).
@misc{sondow2009ramanujan,
abstract = {The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if
$x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For
example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists
and gave the first five values as 2, 11, 17, 29, 41. In this note, we use
inequalities of Rosser and Schoenfeld to prove that $2n \log 2n < R_n < 4n \log
4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is
asymptotic to the $2n$th prime. We also estimate the length of the longest
string of consecutive Ramanujan primes among the first $n$ primes, explain why
there are more twin Ramanujan primes than expected, and make three conjectures
(the first has since been proved by S. Laishram).},
added-at = {2013-12-23T07:00:51.000+0100},
author = {Sondow, Jonathan},
biburl = {https://www.bibsonomy.org/bibtex/27ca0b8e1d79126cac0435dcdd6deb629/aeu_research},
description = {Ramanujan Primes and Bertrand's Postulate},
interhash = {5ca5409f3a0c85e59e8382958199a116},
intrahash = {7ca0b8e1d79126cac0435dcdd6deb629},
keywords = {postulate ramanujan mathematics},
note = {cite arxiv:0907.5232Comment: 7 pages, cited Shapiro's book for Ramanujan's proof of Bertrand's Postulate},
timestamp = {2013-12-24T01:10:11.000+0100},
title = {Ramanujan Primes and Bertrand's Postulate},
url = {http://arxiv.org/abs/0907.5232},
year = 2009
}