The Ramanujan Journal

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

On generalized Ramanujan primes



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.


  1. 1.
    Amersi, N., Beckwith, O., Miller, S.J., Ronan, R., Sondow, J.: Generalized Ramanujan Primes, Combinatorial and Additive Number Theory. Proceedings in Mathematics & Statistics, CANT 2011 and 2012, vol. 101, pp. 1–13. Springer, New York (2014)Google Scholar
  2. 2.
    Axler, C.: New bounds for the prime counting function \(\pi (x)\). arXiv:1409.1780 (2014)
  3. 3.
    Dusart, P.: Autour de la fonction qui compte le nombre de nombres premiers. Dissertation, Université de Limoges (1998)Google Scholar
  4. 4.
    Dusart, P.: The \(k\)th prime is greater than \(k(\ln k+\ln \ln k - 1)\) for \(k\ge 2\). Math. Comput. 68, 411–415 (1999)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dusart, P.: Estimates of some functions over primes without R.H. arXiv:1002.0442v1 (2010)
  6. 6.
    Erdös, P.: Beweis eines Satzes von Tschebyschef. Acta Litt. Sci. Szeged 5, 194–198 (1932)Google Scholar
  7. 7.
    Ishikawa, H.: Über die Verteilung der Primzahlen. Sci. Rep. Tokyo Bunrika Daigaku 2, 27–40 (1934)Google Scholar
  8. 8.
    Laishram, S.: On a conjecture on Ramanujan primes. Int. J. Number Theory 6, 1869–1873 (2010)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Mitra, A., Paul, G., Sarkar, U.: Some conjectures on the number of primes in certain intervals. arXiv:0906.0104v1 (2009)
  10. 10.
    Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Nicholson, J.W.: Sequence A214934, The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015
  12. 12.
    Panaitopol, L.: A formula for \(\pi (x)\) applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. 5(1), 55–56 (2000)MathSciNetGoogle Scholar
  13. 13.
    Ramanujan, S.: A proof of Bertrand’s postulate. J. Indian Math. Soc. 11, 181–182 (1919)Google Scholar
  14. 14.
    Rosser, J.B.: The \(n\)-th prime is greater than \(n \log n\). Proc. Lond. Math. Soc. 45, 21–44 (1939)CrossRefGoogle Scholar
  15. 15.
    Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)MATHMathSciNetGoogle Scholar
  16. 16.
    Shevelev, V.: Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012), Article 12.1.1Google Scholar
  17. 17.
    Sondow, J.: Ramanujan primes and Bertrand’s Postulate. Am. Math. Monthly 116, 630–635 (2009)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Sondow, J.: Sequence A104272. The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015
  19. 19.
    Sondow, J.: Sequence A233739. The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015
  20. 20.
    Sondow, J., Nicholson, J.W., Noe, T.D.: Ramanujan primes: Bounds, Runs, Twins, and Gaps. J. Integer Seq. 14 (2011), Article 11.6.2Google Scholar
  21. 21.
    Srinivasan, A.: An upper bound for Ramanujan primes. Integers 19 (2014), #A19Google Scholar
  22. 22.
    Tchebychev, P.: Mémoire sur les nombres premiers. Mémoires des savants étrangers de l’Acad. Sci. St.Pétersbourg 7 (1850), 17–33 [Also, Journal de mathématiques pures et appliques 17 (1852), 366–390]Google Scholar
  23. 23.
    Trost, E.: Primzahlen. Birkhäuser, Basel/Stuttgart (1953)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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