On Robin’s criterion for the Riemann Hypothesis

Safia Aoudjit, Djamel Berkane and Pierre Dusart
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 15—24
DOI: 10.7546/nntdm.2021.27.4.15-24
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Authors and affiliations

Safia Aoudjit
LAMDA-RO Laboratory, Department of Mathematics, University of Blida1
Po. Box 270 Route de Soumaa, Blida, Algeria

Djamel Berkane
LAMDA-RO Laboratory, Department of Mathematics, University of Blida1
Po. Box 270 Route de Soumaa, Blida, Algeria

Pierre Dusart
Faculté des sciences et techniques, Université de Limoges
P.O. Box 123, avenue Albert Thomas 87060 Limoges Cedex, France

Abstract

Robin’s criterion says that the Riemann Hypothesis is equivalent to

    \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\]

where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates:

    \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*}

The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.

Keywords

  • Primorial number
  • Robin’s inequality
    Riemann Hypothesis

2020 Mathematics Subject Classification

  • 11A25
  • 11N64
  • 11M26

References

  1. Akbary, A., & Francis, F. J. (2020). Euler’s function on products of primes in a fixed arithmetic progression. Mathematics of Computation, 89(322), 993–1026.
  2. Akbary, A., & Friggstad, Z. (2009). Superabundant numbers and the Riemann Hypothesis. American Mathematical Monthly, 116(3), 273–275.
  3. Aoudjit, S., Berkane, D., & Dusart, P. (2021). Explicit estimates involving the primorial integers and applications. Journal of Integer Sequences, 24(7), Article 21.7.8.
  4. Balazard, M. (1990). Unimodalite de la distribution du nombre de diviseurs premiers d’un entier. Annales de l’Institut Fourier, 2, 255–270.
  5. Briggs, K. (2006). Abundant numbers and the Riemann Hypothesis. Experimental Mathematics, 15 (2), 251–256.
  6. Caveney, G., Nicolas, J. L., & Sondow, J. (2011). Robin’s theorem, primes, and a new elementary reformulation of the Riemann Hypothesis. Integers, 11(6), 753–763.
  7. Dusart, P. (2018). Estimates of the k-th prime under the Riemann hypothesis. The Ramanujan Journal, 47(1), 141–154.
  8. Gronwall, T. H. (1913). Some asymptotic expressions in the theory of numbers. Transactions of the American Mathematical Society, 14, 113–122.
  9. Hardy, G. H., & Wright, E. M. (1960). An Introduction to the Theory of Numbers. 4th ed. Oxford University Press.
  10. Lagarias, J. C. (2002). An Elementary Problem Equivalent to the Riemann Hypothesis. American Mathematical Monthly, 109, 534–543.
  11. Nicolas, J. L. (1983). Petites valeurs de la fonction d’Euler. Journal of Number Theory, 17, 375–388.
  12. Robin, G. (1983). Estimation de la fonction de Tchebychef θ sur le kieme nombre premier et grande valeurs de la fonction ω(n) nombre de diviseurs premiers de n. Acta Arithmetica, 42, 367–389.
  13. Robin, G. (1984). Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann. Journal de Mathematiques Pures et Appliqu ees. Neuvieme Serie, 63, 187–213.
  14. Rosser, J. B., & Schoenfeld, L. (1962). Approximate formula for the some functions of Prime Numbers. Illinois Journal of Mathematics, 6, 64–94.
  15. Sloane, N. J. A. A004490. The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org/A004490.

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Cite this paper

Aoudjit, S., Berkane. D., & Dusart, P. (2021). On Robin’s criterion for the Riemann Hypothesis. Notes on Number Theory and Discrete Mathematics, 27(4), 15-24, doi: 10.7546/nntdm.2021.27.4.15-24.

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