Dickman-de Bruijn function
History
The function was first introduced by Dickman with a heuristic argument relating it to smoothness. de Bruijn explored many properties of this function, and Ramaswami gave a formal proof of its relation to the size of the largest prime divisor.
Definition
This function, called Dickman's function or the Dickman-de Bruijn function, is defined as the function satisfying the delay differential equation:
subject to the initial condition for
. The function satisfies the following properties:
-
for
.
-
for
.
-
is (strictly) decreasing for
, i.e.,
for
.
-
is once differentiable on
. More generally,
is
times differentiable everywhere except at the points
.
-
is infinitely differentiable except at integers.
-
.
It turns out that the density of numbers with no prime divisor greater than the root is given by
. Formally, consider, for any
, the fraction of natural numbers
such that all prime divisors of
are at most
. Then, as
, this fraction tends to
. Thus, this function is crucial to understand the behavior of the largest prime divisor function and it is important in obtaining smoothness bounds.