Fungrim entry: 9697b8

\log\!\left(g(n)\right) \le \sqrt{n \log(n)} \left(1 + \frac{\log\!\left(\log(n)\right) - 0.975}{2 \log(n)}\right)

Assumptions:

n \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(n)\right) \le \sqrt{n \log(n)} \left(1 + \frac{\log\!\left(\log(n)\right) - 0.975}{2 \log(n)}\right)

n \in \mathbb{Z}_{\ge 4}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`LandauG`	$g(n)$	Landau's function
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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"))

Topics using this entry

Landau's function