On certain inequalities for the prime counting function – Part III

József Sándor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 454–461
DOI: 10.7546/nntdm.2023.29.3.454-461
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Authors and affiliations

József Sándor
Department of Mathematics, Babeș-Bolyai University
Cluj-Napoca, Romania

Abstract

As a continuation of [10] and [11], we offer some new inequalities for the prime counting function \pi (x). Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau’s inequality are given. Some results on \pi (p_n^2) are offered, p_n denoting the n-th prime number. Results on \pi (\pi (x)) are also considered.

Keywords

  • Prime counting function
  • Inequalities
  • Hardy–Littlewood conjecture
  • Landau’s inequality
  • Prime numbers

2020 Mathematics Subject Classification

  • 11A25
  • 11A41

References

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  2. Alzer, H., Kwong, M. K., & Sándor, J. (2022). Inequalities involving \pi(x). Rendiconti del Seminario Matematico della Università di Padova, 147(1), 237–251.
  3. Dusart, P. (1999). Inégalités explicit pour \psi(x), \theta (x), \pi(x) et les nombres premiers. Comptes rendus mathématiques de l’Académie des sciences. La Société Royale du Canada, 21(2), 53–59.
  4. Miliakos, G. (2022). Open Questions OQ 5762 and OQ 5760. Octogon Mathematical Magazine, 30(2), 1510.
  5. Panaitopol, L. (1998). On the inequality \pi(x) > \frac{x}{\log x-1}. Analele Universităţii din Bucureşti, Seria Matematica. XLVII(2), 187–192.
  6. Panaitopol, L. (1998). On the inequality p_a p_b > p_{ab}. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 41(89), No. 2, 135–139.
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  8. Rosser, J. B., & Schoenfeld, L. (1962). Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6, 64–94.
  9. Sándor, J. (2006). On some inequalities of Dusart and Panaitopol on the function π(x). Octogon Mathematical Magazine, 14 (2), 592–594.
  10. Sándor, J. (2021). On certain inequalities for the prime counting function. Notes on Number Theory and Discrete Mathematics, 27(4), 149–153.
  11. Sándor, J. (2022). On certain inequalities for the prime counting function – Part II. Notes on Number Theory and Discrete Mathematics, 28(1), 124–128.
  12. Sándor, J., Mitrinović, D. S., & Cristici, B. (2005). Handbook of Number Theory I, Springer.

Manuscript history

  • Received: 5 May 2023
  • Revised: 21 June 2023
  • Accepted: 30 June 2023
  • Online First: 3 July 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Sándor, J. (2023). On certain inequalities for the prime counting function – Part III. Notes on Number Theory and Discrete Mathematics, 29(3), 454-461, DOI: 10.7546/nntdm.2023.29.3.454-461.

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