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


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.


  • Primorial number
  • Robin’s inequality
    Riemann Hypothesis

2020 Mathematics Subject Classification

  • 11A25
  • 11N64
  • 11M26


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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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