Dickman function

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.¹ It was later studied by the Dutch mathematician Nicolaas Govert de Bruijn.²³

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

${\displaystyle u\rho '(u)+\rho (u-1)=0\,}$

with initial conditions ${\displaystyle \rho (u)=1}$ for 0 ≤ u ≤ 1.

Dickman proved that, when ${\displaystyle a}$ is fixed, we have

${\displaystyle \Psi (x,x^{1/a})\sim x\rho (a)\,}$

where ${\displaystyle \Psi (x,y)}$ is the number of y-smooth (or y-friable) integers below x. Equivalently, the number of ${\displaystyle B}$-smooth numbers less than ${\displaystyle N}$ is about $${\displaystyle \Psi (N,B)\approx N\rho \left({\frac {\log N}{\log B}}\right).}$$

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

${\displaystyle \Psi (x,x^{1/a})=x\rho (a)+O(x/\log x)}$

in big O notation.⁴

Knuth gives a proof for a narrowed bound:

${\displaystyle \Psi (x,x^{1/a})=x\rho (a)+(1-\gamma )\rho (a-1)(x/\log x)+O(x/{(\log x)}^{2})}$

where γ is Euler's constant.^5: 98

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.⁵

It can be shown that⁶

${\displaystyle \Psi (x,y)=xu^{O(-u)}}$

which is related to the estimate ${\displaystyle \rho (u)\approx u^{-u}}$ below.

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

A first approximation might be ${\displaystyle \rho (u)\approx u^{-u}.\,}$ A better estimate is⁷

${\displaystyle \rho (u)\sim {\frac {1}{\xi {\sqrt {2\pi u}}}}\cdot \exp(-u\xi +\operatorname {Ei} (\xi ))}$

where Ei is the exponential integral and ξ is the positive root of

${\displaystyle e^{\xi }-1=u\xi .\,}$

A simple upper bound is ${\displaystyle \rho (x)\leq 1/x!.}$

${\displaystyle u}$	${\displaystyle \rho (u)}$
1	1
2	3.0685282×10−1
3	4.8608388×10−2
4	4.9109256×10−3
5	3.5472470×10−4
6	1.9649696×10−5
7	8.7456700×10−7
8	3.2320693×10−8
9	1.0162483×10−9
10	2.7701718×10−11

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

${\displaystyle \rho (u)=1-(1-\log(u-1))\log(u)+\operatorname {Li} _{2}(1-u)+{\frac {\pi ^{2}}{12}}.}$

with Li₂ the dilogarithm. Other ${\displaystyle \rho _{n}}$ can be calculated using infinite series.⁸

An alternate method is computing lower and upper bounds with the trapezoidal rule;⁷ 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.⁹ Values for u ≤ 7 can be usefully computed via numerical integration in ordinary double-precision floating-point.^5: 99

Friedlander defines a two-dimensional analog ${\displaystyle \sigma (u,v)}$ of ${\displaystyle \rho (u)}$.¹⁰ This function is used to estimate a function ${\displaystyle \Psi (x,y,z)}$ similar to de Bruijn's, but counting the number of y-smooth integers with at most one prime factor greater than z. Then

${\displaystyle \Psi (x,x^{1/a},x^{1/b})\sim x\sigma (b,a).\,}$

This class of numbers may be encountered in the two-stage variant of P-1 factoring. However, Kruppa's estimate of the probability of finding a factor by P-1 does not make use of this result.^5: 100