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.

V. Ramaswami of Andhra University 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).\,

See also

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" (PDF). Indagationes Mathematicae 13: 50–60.
  3. ↑ 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.
  4. ↑ Ramaswami, V. (1949). x (PDF). Bulletin of the American Mathematical Society 55: 1122–1127. doi:10.1090/s0002-9904-1949-09337-0.
  5. ↑ 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.
  6. ↑ 6.0 6.1 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. ↑ Bach, Eric; Peralta, René (1996). "Asymptotic Semismoothness Probabilities" (PDF). Mathematics of Computation 65 (216): 1701–1715. doi:10.1090/S0025-5718-96-00775-2.
  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.
  9. ↑ 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.

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