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 $\rho(u)$ 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. 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)

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 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 $\rho(x)\le1/x!.$

$u$	$\rho(u)$
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) + \operatorname{Li}_2(1 - u) + \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

Friedlander defines a two-dimensional analog $\sigma(u,v)$ of $\rho(u)$ .^[9] 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

↑ 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.
↑ de Bruijn, N. G. (1951). "On the number of positive integers ≤ x and free of prime factors > y" (PDF). Indagationes Mathematicae 13: 50–60.
↑ de Bruijn, N. G. (1966). "On the number of positive integers ≤ x and free of prime factors > y, II" (PDF). Indagationes Mathematicae 28: 239–247.
↑ Ramaswami, V. (1949). $x$ (PDF). Bulletin of the American Mathematical Society 55 (12): 1122–1127. doi:10.1090/s0002-9904-1949-09337-0. MR 0031958.
↑ Hildebrand, A.; Tenenbaum, G. (1993). "Integers without large prime factors" (PDF). Journal de théorie des nombres de Bordeaux 5 (2): 411–484. doi:10.5802/jtnb.101.
1 2 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.
↑ Bach, Eric; Peralta, René (1996). "Asymptotic Semismoothness Probabilities" (PDF). Mathematics of Computation 65 (216): 1701–1715. doi:10.1090/S0025-5718-96-00775-2.
↑ 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.
↑ Friedlander, John B. (1976). "Integers free from large and small primes". Proc. London Math. Soc. 33 (3): 565–576. doi:10.1112/plms/s3-33.3.565.