# Dickman-de Bruijn function

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## 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.