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.
