Dickman-de Bruijn function
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.
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.