In analytic number theory, Dickman's function is a special function used to estimate the proportion of smooth numbers up to a given bound.

Dickman's function is named after actuary Karl Dickman, who defined it in his only mathematical publication. Its properties were later studied by the Dutch mathematician Nicolaas Govert de Bruijn;, so some sources call it the Dickman-de Bruijn function.

Definition

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

$urho\text{'}(u)\; +\; rho(u-1)\; =\; 0,$

with initial conditions $rho(u)\; =\; 1$ for 0 ≤ $u$ ≤ 1. Dickman showed heuristically that

$Psi(x,\; x^\{1/a\})sim\; xrho(a),$

where $Psi(x,y)$ is the number of y-smooth integers below x.

V. Ramaswami of Andhra University later gave a rigorous proof that $Psi(x,x^\{1/a\})$ was asymptotic to $rho(a)$, along with the error bound

$Psi(x,x^\{1/a\})=xrho(a)+mathcal\{O\}(x/log\; x)$

in big O notation.

Applications

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, and can be useful of its own right.

It can be shown using $logrho$ 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

$rho(u)simfrac\{1\}\{xisqrt\{2pi\; x\}\}cdotexp(-xxi+operatorname\{Ei\}(xi))$

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

$e^xi-1=xxi.,$

A simple upper bound is $rho(x)le1/x!.$

$u$	$rho(u)$
1	1
2	3.0685282
3	4.8608388
4	4.9109256
5	3.5472470
6	1.9649696
7	8.7456700
8	3.2320693
9	1.0162483
10	2.7701718

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

Bach and Peralta define 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

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

References

External links