Ramanujan Primes: Bounds, Runs, Twins, and Gaps

TL;DR

This paper investigates properties of Ramanujan primes, providing bounds, statistical analysis of their runs, relationships with twin primes, and connections to prime gaps, along with conjectures and computational methods.

Contribution

It refines bounds on Ramanujan primes, analyzes their distribution and runs, explores their relation to twin primes, and links them to prime gaps, including new conjectures and algorithms.

Findings

Maximum of R_n/p_{3n} is R_5/p_{15} = 41/47

Longest runs of Ramanujan primes among primes less than 10^9 analyzed

If an upper twin prime is Ramanujan, then the lower twin prime is also Ramanujan

Abstract

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. The Appendix explains Noe's fast algorithm for computing $R_1,R_2,...,R_n$.