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September 2020 On Ramanujan primes
Christian Axler
Funct. Approx. Comment. Math. 63(1): 67-93 (September 2020). DOI: 10.7169/facm/1824

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 $(\pi(R_n))_{n \geq 1}$, 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. Then we use these explicit estimates to derive a result concerning an inequality involving $\pi(R_n)$ that was conjectured by Sondow, Nicholson, and Noe. In the second part of the paper, we apply the results derived in the first part to obtain some new results concerning the number of Ramanujan primes not exceeding $x$. Finally, we compute the limit of $((R_n - p_{2n})/n)_{n \geq 1}$.

Citation

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Christian Axler. "On Ramanujan primes." Funct. Approx. Comment. Math. 63 (1) 67 - 93, September 2020. https://doi.org/10.7169/facm/1824

Information

Published: September 2020
First available in Project Euclid: 14 December 2019

MathSciNet: MR4149511
Digital Object Identifier: 10.7169/facm/1824

Subjects:
Primary: 11N05
Secondary: 11A41

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.63 • No. 1 • September 2020
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