- Landau's function
Landau's function "g"("n") is defined for every
natural number "n" to be the largest order of an element of thesymmetric group "S""n". Equivalently, "g"("n") is the largestleast 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 "S"5 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 [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793 A000793] .The sequence is named after
Edmund Landau , who proved in 1902 (reference [1] below) that:lim_{n oinfty}frac{ln(g(n))}{sqrt{n ln(n) = 1(where ln denotes thenatural logarithm ).The statement that :ln g(n)
for all n, where Li -1 denotes the inverse of thelogarithmic integral function , is equivalent to theRiemann hypothesis .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.External links
On-Line Encyclopedia of Integer Sequences : Sequence [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793 A000793,] Landau's function on the natural numbers.
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