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

TL;DR

This paper investigates the distribution of Ramanujan primes, establishing new bounds for the prime count up to the $n$th Ramanujan prime and confirming related conjectures about their asymptotic behavior.

Contribution

The paper provides explicit bounds for the number of primes up to the $n$th Ramanujan prime and proves an asymptotic formula conjectured by Yang and T"oge.

Findings

Established new explicit upper and lower bounds for $\, ext{pi}(R_n)$

Derived an asymptotic formula for $ ext{pi}(R_n)$

Confirmed a conjectured inequality involving $ ext{pi}(R_n)$

Abstract

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ 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 Togb\'e. 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.