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The Ramanujan Journal

, Volume 39, Issue 1, pp 1–30 | Cite as

On generalized Ramanujan primes

  • Christian Axler
Article

Abstract

In this paper, we establish several results concerning the generalized Ramanujan primes. For \(n\in \mathbb {N}\) and \(k \in \mathbb {R}_{> 1}\), we give estimates for the \(n\)th \(k\)-Ramanujan prime, which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of \(k\)-Ramanujan primes. In addition, we find explicit formulae for certain \(n\)th \(k\)-Ramanujan primes. As an application, we prove that a conjecture of Mitra et al. (arXiv:0906.0104v1, 2009) concerning the number of primes in certain intervals holds for every sufficiently large positive integer.

Keywords

Ramanujan primes Bertrand’s postulate Distribution of prime numbers 

Mathematics Subject Classification

11N05 11A41 11B05 

Notes

Acknowledgments

I would like to thank Benjamin Klopsch for the helpful conversations. Also I would like to thank Elena Klimenko and Anitha Thillaisundaram for their careful reading of the paper.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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