Landau's function

In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group S_n. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence.

For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6. An element of order 6 in the group S₅ can be written in cycle notation as (1 2) (3 4 5).

The integer sequence g(0) = 1, g(1) = 1, g(2) = 2, g(3) = 3, g(4) = 4, g(5) = 6, g(6) = 6, g(7) = 12, g(8) = 15, ... (sequence A000793 in OEIS) is named after Edmund Landau, who proved in 1902^[1] that

$\lim_{n\to\infty}\frac{\ln(g(n))}{\sqrt{n \ln(n)}} = 1$

(where ln denotes the natural logarithm).

The statement that

$\ln g(n)<\sqrt{\mathrm{Li}^{-1}(n)}$

for all sufficiently large n, where Li⁻¹ denotes the inverse of the logarithmic integral function, is equivalent to the Riemann hypothesis.

It can be shown that

$g(n)<e^{n/e}$

and indeed

$g(n) \le \exp\left(1.05314\sqrt{n\log n}\right).$^[2]

Notes

1. Landau, pp. 92–103
2. Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, pp. 269–281 (1985).

References

• E. Landau, "Über die Maximalordnung der Permutationen gegebenen Grades [On the maximal order of permutations of given degree]", Arch. Math. Phys. Ser. 3, vol. 5, 1903.
• W. Miller, "The maximum order of an element of a finite symmetric group", American Mathematical Monthly, vol. 94, 1987, pp. 497–506.
• J.-L. Nicolas, "On Landau's function g(n)", in The Mathematics of Paul Erdős, vol. 1, Springer-Verlag, 1997, pp. 228–240.

External link

• "Sloane's A000793 : Landau's function on the natural numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.