Landau's function

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Landau's function g(n) 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 of any partition of n.

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, ... is A000793.

The sequence is named after Edmund Landau, who proved in 1902 (reference [1] below) 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 n, where Li^-1 denotes the inverse of the logarithmic integral function, is equivalent to the Riemann hypothesis.

[edit] 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, pp. 92-103.
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.