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 $\rho :[0,\infty )\to \mathbb {R}$ satisfying the delay differential equation:

$\!u\rho '(u)+\rho (u-1)=0$

subject to the initial condition $\rho (u)=1$ for $u\in [0,1]$. The function satisfies the following properties:

• $\rho (u)=1$ for $u\in [0,1]$.
• $\rho (u)=1-\log u$ for $u\in [1,2]$.
• $\rho$ is (strictly) decreasing for $u\in [1,\infty )$, i.e., $\rho '(u)<0$ for $u\in (1,\infty )$.
• $\rho$ is once differentiable on $(1,\infty )$. More generally, $\rho$ is $n$ times differentiable everywhere except at the points $\{1,2,\dots ,n\}$.
• $\rho$ is infinitely differentiable except at integers.
• $\lim _{u\to \infty }\rho (u)=0$.

It turns out that the density of numbers with no prime divisor greater than the $x^{th}$ root is given by $\rho (x)$. Formally, consider, for any $N$, the fraction of natural numbers $n\leq N$ such that all prime divisors of $n$ are at most $N^{1/x}$. Then, as $N\to \infty$, this fraction tends to $\rho (x)$. Thus, this function is crucial to understand the behavior of the largest prime divisor function and it is important in obtaining smoothness bounds.