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Fungrim entry: 9697b8

log ⁣(g ⁣(n))nlog ⁣(n)(1+log ⁣(log ⁣(n))0.9752log ⁣(n))\log\!\left(g\!\left(n\right)\right) \le \sqrt{n \log\!\left(n\right)} \left(1 + \frac{\log\!\left(\log\!\left(n\right)\right) - 0.975}{2 \log\!\left(n\right)}\right)
Assumptions:nZ4n \in \mathbb{Z}_{\ge 4}
References:
  • Jean-Pierre Massias, Jean-Louis Nicolas and Guy Robin (1989), Effective bounds for the maximal order of an element in the symmetric group, Mathematics of Computation, 53, 118, 665--665, https://doi.org/10.1090/s0025-5718-1989-0979940-4
TeX:
\log\!\left(g\!\left(n\right)\right) \le \sqrt{n \log\!\left(n\right)} \left(1 + \frac{\log\!\left(\log\!\left(n\right)\right) - 0.975}{2 \log\!\left(n\right)}\right)

n \in \mathbb{Z}_{\ge 4}
Definitions:
Fungrim symbol Notation Short description
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
LandauGg ⁣(n)g\!\left(n\right) Landau's function
Sqrtz\sqrt{z} Principal square root
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("9697b8"),
    Formula(LessEqual(Log(LandauG(n)), Mul(Sqrt(Mul(n, Log(n))), Add(1, Div(Sub(Log(Log(n)), Decimal("0.975")), Mul(2, Log(n))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(4))),
    References("Jean-Pierre Massias, Jean-Louis Nicolas and Guy Robin (1989), Effective bounds for the maximal order of an element in the symmetric group, Mathematics of Computation, 53, 118, 665--665, https://doi.org/10.1090/s0025-5718-1989-0979940-4"))

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2019-08-21 11:44:15.926409 UTC