# Dickman function The Dickman–de Bruijn function ρ(u) plotted on a logarithmic scale. The horizontal axis is the argument u, and the vertical axis is the value of the function. The graph nearly makes a downward line on the logarithmic scale, demonstrating that the logarithm of the function is quasilinear.

In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound. It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, and later studied by the Dutch mathematician Nicolaas Govert de Bruijn.

## Definition

The Dickman–de Bruijn function $\rho (u)$ is a continuous function that satisfies the delay differential equation

$u\rho '(u)+\rho (u-1)=0\,$ with initial conditions $\rho (u)=1$ for 0 ≤ u ≤ 1.

## Properties

Dickman proved that, when $a$ is fixed, we have

$\Psi (x,x^{1/a})\sim x\rho (a)\,$ where $\Psi (x,y)$ is the number of y-smooth (or y-friable) integers below x.

Ramaswami later gave a rigorous proof that for fixed a, $\Psi (x,x^{1/a})$ was asymptotic to $x\rho (a)$ , with the error bound

$\Psi (x,x^{1/a})=x\rho (a)+O(x/\log x)$ ## Applications The Dickman–de Bruijn used to calculate the probability that the largest and 2nd largest factor of x is less than x^a

The main purpose of the Dickman–de Bruijn function is to estimate the frequency of smooth numbers at a given size. This can be used to optimize various number-theoretical algorithms such as P-1 factoring and can be useful of its own right.

It can be shown using $\log \rho$ that

$\Psi (x,y)=xu^{O(-u)}$ which is related to the estimate $\rho (u)\approx u^{-u}$ below.

The Golomb–Dickman constant has an alternate definition in terms of the Dickman–de Bruijn function.

## Estimation

A first approximation might be $\rho (u)\approx u^{-u}.\,$ A better estimate is

$\rho (u)\sim {\frac {1}{\xi {\sqrt {2\pi u}}}}\cdot \exp(-u\xi +\operatorname {Ei} (\xi ))$ where Ei is the exponential integral and ξ is the positive root of

$e^{\xi }-1=u\xi .\,$ A simple upper bound is $\rho (x)\leq 1/x!.$ $u$ $\rho (u)$ 1 1
2 3.0685282×101
3 4.8608388×102
4 4.9109256×103
5 3.5472470×104
6 1.9649696×105
7 8.7456700×107
8 3.2320693×108
9 1.0162483×109
10 2.7701718×1011

## Computation

For each interval [n − 1, n] with n an integer, there is an analytic function $\rho _{n}$ such that $\rho _{n}(u)=\rho (u)$ . For 0 ≤ u ≤ 1, $\rho (u)=1$ . For 1 ≤ u ≤ 2, $\rho (u)=1-\log u$ . For 2 ≤ u ≤ 3,

$\rho (u)=1-(1-\log(u-1))\log(u)+\operatorname {Li} _{2}(1-u)+{\frac {\pi ^{2}}{12}}.$ with Li2 the dilogarithm. Other $\rho _{n}$ can be calculated using infinite series.

An alternate method is computing lower and upper bounds with the trapezoidal rule; a mesh of progressively finer sizes allows for arbitrary accuracy. For high precision calculations (hundreds of digits), a recursive series expansion about the midpoints of the intervals is superior.

## Extension

Friedlander defines a two-dimensional analog $\sigma (u,v)$ of $\rho (u)$ . This function is used to estimate a function $\Psi (x,y,z)$ similar to de Bruijn's, but counting the number of y-smooth integers with at most one prime factor greater than z. Then

$\Psi (x,x^{1/a},x^{1/b})\sim x\sigma (b,a).\,$ • Buchstab function, a function used similarly to estimate the number of rough numbers, whose convergence to $e^{-\gamma }$ is controlled by the Dickman function