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,^[1] and later studied by the Dutch mathematician Nicolaas Govert de Bruijn.^[2][3]

Definition

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

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

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

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

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

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

Ψ(x,x^{1 / a}) = xρ(a) + O(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 ρ that^[5]

Ψ(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]

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

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	ρ(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 ρ_n such that ρ_n(u) = ρ(u). For 0 ≤ u ≤ 1, ρ(u) = 1. For 1 ≤ u ≤ 2, ρ(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 ρ_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 σ(u,v) of ρ(u).^[7] This function is used to estimate a function Ψ(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 x^c". 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

• Broadhurst, David (2010). "Dickman polylogarithms and their constants". arXiv:1004.0519. 
• Soundararajan, K. (2010). "An asymptotic expansion related to the Dickman function". arXiv:1005.3494. 
• Weisstein, Eric W., "Dickman function" from MathWorld.