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<sub>n</sub>. 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 S<sub>5</sub> can be written in cycle notation as (1 2) (3 4 5). Note that the same argument applies to the number 6, that is, g(6) = 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(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 named after Edmund Landau, who proved in 1902 that
(where ln denotes the natural logarithm). Equivalently (using little-o notation), .
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