Ramanujan Primes and Bertrand's Postulate

TL;DR

This paper investigates properties of Ramanujan primes, providing bounds, asymptotic behavior, and conjectures, thereby deepening understanding of their distribution and relation to prime numbers.

Contribution

It establishes bounds for Ramanujan primes using inequalities, proves their asymptotic equivalence to the 2n-th prime, and explores their distribution and conjectures.

Findings

Bounds: 2n log 2n < R_n < 4n log 4n

Asymptotic to the 2n-th prime

More twin Ramanujan primes than expected

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).