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On Landau’s Function g(n)

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The Mathematics of Paul Erdős I

Abstract

Let S n be the symmetric group of n letters. Landau considered the function g(n) defined as the maximal order of an element of S n ; Landau observed that (cf. [9])

$$\displaystyle{ g(n) =\max \mathrm{lcm}(m_{1},\ldots,m_{k}) }$$
(1)

where the maximum is taken on all the partitions \(n = m_{1} + m_{2} + \cdots + m_{k}\) of n and proved that, when n tends to infinity

$$\displaystyle{ \log g(n) \sim \sqrt{n\log n}. }$$
(2)

More precise asymptotic estimates have been given in [11, 22, 25].

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Authors and Affiliations

  1. Institut Camille Jordan, Mathématiques, Université de Lyon, CNRS, Université Lyon 1, 43 Bd. du 11 Novembre 1918, F-69622, Villeurbanne Cedex, France

    Jean-Louis Nicolas

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  1. Jean-Louis Nicolas
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Correspondence to Jean-Louis Nicolas .

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Editors and Affiliations

  1. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University Department of Applied Mathematics, Prague, Czech Republic

    Jaroslav Nešetřil

  3. Department of Mathematics, Iowa State University, Ames, Iowa, USA

    Steve Butler

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Nicolas, JL. (2013). On Landau’s Function g(n). In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_14

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