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 Sn. 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)[citation needed]

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.