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).
Users
Please
log in to take part in the discussion (add own reviews or comments).