Ramanujan Primes: Bounds, Runs, Twins, and Gaps

The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, then the interval ( 1 2x, x ] contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below 10n of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. An appendix explains Noe’s fast algorithm for computing R1, R2, . . . , Rn.