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.
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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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Axler, C. On generalized Ramanujan primes. Ramanujan J 39, 1–30 (2016). https://doi.org/10.1007/s11139-015-9693-9
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DOI: https://doi.org/10.1007/s11139-015-9693-9