Proceedings of the Japan Academy, Series A, Mathematical Sciences

Robin's inequality and the Riemann hypothesis

Marek Wójtowicz

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Abstract

Let $f(n)=\sigma(n)/e^\gamma n\log\log n$, $n=3,4,\ldots$ , where $\sigma$ denotes the sum of divisors function. In 1984 Robin proved that the inequality $f(n)>1$, for all $n\ge 5041$, is equivalent to the Riemann hypothesis. Here we show that the values of $f$ are close to $0$ on a set of asymptotic density $1$. Similarly, an inequality by Rosser and Schoenfeld of 1962, dealing with the Euler totient function $\varphi$, is essential only on "thin" subsets of $\mathbf{N}$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 83, Number 4 (2007), 47-49.

Dates
First available in Project Euclid: 30 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.pja/1177941416

Digital Object Identifier
doi:10.3792/pjaa.83.47

Mathematical Reviews number (MathSciNet)
MR2326201

Zentralblatt MATH identifier
1142.11064

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11N37: Asymptotic results on arithmetic functions

Keywords
Riemann hypothesis Robin's inequality asymptotic density

Citation

Wójtowicz, Marek. Robin's inequality and the Riemann hypothesis. Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 4, 47--49. doi:10.3792/pjaa.83.47. https://projecteuclid.org/euclid.pja/1177941416


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References

  • L. Alaoglu and P. Erdös, On highly composite and similar numbers, Trans. Amer. Math. Soc. 56 (1944), 448–469.
  • K. Briggs, Abundant numbers and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 2, 251–256.
  • J. H. Bruinier, Primzahlen, Teilersummen und die Riemannsche Vermutung, Math. Semesterber. 48 (2001), no. 1, 79–92.
  • K. Ford, An explicit sieve bound and small values of $\sigma(\phi(m))$, Period. Math. Hungar. 43 (2001), no. 1-2, 15–29.
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fifth edition, Oxford Univ. Press, New York, 1979.
  • J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Amer. Math. Monthly 109 (2002), no. 6, 534–543.
  • F. Luca and C. Pomerance, On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions $\phi$ and $\sigma$, Colloq. Math. 92 (2002), no. 1, 111–130.
  • A. Mąkowski and A. Schinzel, On the functions $\varphi (n)$ and $\sigma (n)$, Colloq. Math. 13 (1964), 95–99.
  • J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17 (1983), no. 3, 375–388.
  • T. Noe, Sequence A073751, published in: The On-Line Encyclopedia of Integer Sequences, 2002. (Electronic).
  • G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. (9) 63 (1984), no. 2, 187–213.
  • J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.
  • W. Sierpiński, Elementary theory of numbers, Translated from Polish by A. Hulanicki. Monografie Matematyczne, Państwowe Wydawnictwo Naukowe, Warsaw, 1964.
  • G. Tenenbaum, Introduction to analytic and probabilistic number theory, Translated from the second French edition (1995) by C. B. Thomas, Cambridge Univ. Press, Cambridge, 1995.