The Ramanujan Journal

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

On generalized Ramanujan primes

  • Christian Axler


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.


Ramanujan primes Bertrand’s postulate Distribution of prime numbers 

Mathematics Subject Classification

11N05 11A41 11B05 



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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