Dickman-de Bruijn function

Definition

This function, called Dickman's function or the Dickman-de Bruijn function, is defined as the function $\rho :[0,\mathrm{\infty })\to \mathbb{R}$ satisfying the delay differential equation:

$u{\rho }^{\prime }(u)+\rho (u-1)=0$

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 $u\in [1,\mathrm{\infty })$, i.e., ${\rho }^{\prime }(u)<0$ for $u\in (1,\mathrm{\infty })$.
• $\rho$ is once differentiable on $(1,\mathrm{\infty })$. More generally, $\rho</math> is $n$ times differentiable everywhere except at the points $\{1,2,\dots ,n\}$.
• $\rho$ is infinitely differentiable except at integers.
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Related facts

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 $n\le N$ such that all prime divisors of $n$ are at most $N^{1/x}$. Then, as $N\to \mathrm{\infty }$, this fraction tends to $\rho (x)$.