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] and later studied by the Dutch mathematician Nicolaas Govert de Bruijn.[2][3]

Contents

Definition

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

u\rho'(u) %2B \rho(u-1) = 0\,

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

\Psi(x, x^{1/a})\sim x\rho(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 x \rho(a), with the error bound

\Psi(x,x^{1/a})=x\rho(a)%2BO(x/\log x)

in big O notation.[4]

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 \log\rho that[5]

\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[6]

\rho(u)\sim\frac{1}{\xi\sqrt{2\pi x}}\cdot\exp(-x\xi%2B\operatorname{Ei}(\xi))

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

e^\xi-1=x\xi.\,

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

u \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

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) %2B \operatorname{Li}_2(1 - u) %2B \frac{\pi^2}{12}.

with Li2 the dilogarithm. Other \rho_n can be calculated using infinite series.[7]

An alternate method is computing lower and upper bounds with the trapezoidal rule;[6] 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.[8]

Extension

Bach and Peralta define a two-dimensional analog \sigma(u,v) of \rho(u).[7] 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).\,

References

  1. ^ Dickman, K. (1930). "On the frequency of numbers containing prime factors of a certain relative magnitude". Arkiv för Matematik, Astronomi och Fysik 22A (10): 1–14. 
  2. ^ de Bruijn, N. G. (1951). "On the number of positive integers ≤ x and free of prime factors > y". Indagationes Mathematicae 13: 50–60. http://alexandria.tue.nl/repository/freearticles/597499.pdf. 
  3. ^ de Bruijn, N. G. (1966). "On the number of positive integers ≤ x and free of prime factors > y, II". Indagationes Mathematicae 28: 239–247. http://alexandria.tue.nl/repository/freearticles/597534.pdf. 
  4. ^ Ramaswami, V. (1949). "On the number of positive integers less than x and free of prime divisors greater than xc". Bulletin of the American Mathematical Society 55: 1122–1127. http://www.ams.org/bull/1949-55-12/S0002-9904-1949-09337-0/S0002-9904-1949-09337-0.pdf. 
  5. ^ Hildebrand, A.; Tenenbaum, G. (1993). "Integers without large prime factors". Journal de théorie des nombres de Bordeaux 5 (2): 411–484. http://archive.numdam.org/article/JTNB_1993__5_2_411_0.pdf. 
  6. ^ a b van de Lune, J.; Wattel, E. (1969). "On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory". Mathematics of Computation 23 (106): 417–421. doi:10.1090/S0025-5718-1969-0247789-3. 
  7. ^ a b Bach, Eric; Peralta, René (1996). "Asymptotic Semismoothness Probabilities". Mathematics of Computation 65 (216): 1701–1715. doi:10.1090/S0025-5718-96-00775-2. http://cr.yp.to/bib/1996/bach-semismooth.pdf. 
  8. ^ Marsaglia, George; Zaman, Arif; Marsaglia, John C. W. (1989). "Numerical Solution of Some Classical Differential-Difference Equations". Mathematics of Computation 53 (187): 191–201. doi:10.1090/S0025-5718-1989-0969490-3. 

External links