Ramanujan Primes and Bertrand's Postulate

@article{Sondow2009RamanujanPA,
  title={Ramanujan Primes and Bertrand's Postulate},
  author={Jonathan Sondow},
  journal={The American Mathematical Monthly},
  year={2009},
  volume={116},
  pages={630 - 635}
}
  • J. Sondow
  • Published 29 July 2009
  • Mathematics
  • The American Mathematical Monthly
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… 
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