Dickman function

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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[edit]

The Dickman–de Bruijn function is a continuous function that satisfies the delay differential equation

with initial conditions for 0 ≤ u ≤ 1. Dickman proved that, when is fixed, we have

where is the number of y-smooth (or y-friable) integers below x.

Ramaswami later gave a rigorous proof that for fixed a, was asymptotic to , with the error bound

in big O notation.[4]

Applications[edit]

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 that[5]

which is related to the estimate below.

The Golomb–Dickman constant has an alternate definition in terms of the Dickman–de Bruijn function.

Estimation[edit]

A first approximation might be A better estimate is[6]

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

A simple upper bound is

1 1
2 3.0685282×101
3 4.8608388×102
4 4.9109256×103
5 3.5472470×104
6 1.9649696×105
7 8.7456700×107
8 3.2320693×108
9 1.0162483×109
10 2.7701718×1011

Computation[edit]

For each interval [n − 1, n] with n an integer, there is an analytic function such that . For 0 ≤ u ≤ 1, . For 1 ≤ u ≤ 2, . For 2 ≤ u ≤ 3,

with Li2 the dilogarithm. Other 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[edit]

Friedlander defines a two-dimensional analog of .[9] This function is used to estimate a function similar to de Bruijn's, but counting the number of y-smooth integers with at most one prime factor greater than z. Then

See also[edit]

References[edit]

  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). "On the number of positive integers less than and free of prime divisors greater than xc" (PDF). Bulletin of the American Mathematical Society. 55 (12): 1122–1127. doi:10.1090/s0002-9904-1949-09337-0. MR 0031958. 
  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. ^ 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. ^ 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. 

Further reading[edit]