On the number of primes up to the $n$th Ramanujan prime

Abstract

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x\ge R_n$ the interval $(x/2,x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi (R_n){)}_{n\in \mathbb{N}}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $\pi (R_n)$ conjectured by Yang and Togbé. In the second part of this paper, we use these explicit estimates to derive a result concerning an inequality involving $\pi (R_n)$ conjectured by of Sondow, Nicholson and Noe.

13 pages

References in corpus (1)

• Estimates of Some Functions Over Primes without R.H.