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.