Landau's function
In mathematics, Landau's function g, 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 is the largest least common multiple 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 = 6. No other partition of 5 yields a bigger lcm, so g = 6. An element of order 6 in the group S5 can be written in cycle notation as . Note that the same argument applies to the number 6, that is, g = 6. There are arbitrarily long sequences of consecutive numbers n, n + 1,..., n + m on which the function g is constant.
The integer sequence g = 1, g = 1, g = 2, g = 3, g = 4, g = 6, g = 6, g = 12, g = 15,... is named after Edmund Landau, who proved in 1902 that
. Equivalently,.
More precisely,
If, where denotes the prime counting function, the logarithmic integral function with inverse, and we may take for some constant c > 0 by Ford, then
The statement that
for all sufficiently large n is equivalent to the Riemann hypothesis.
It can be shown that
with the only equality between the functions at n = 0, and indeed