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词条 Dickman function
释义

  1. Definition

  2. Applications

  3. Estimation

  4. Computation

  5. Extension

  6. See also

  7. References

  8. Further reading

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 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

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

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
2-1}}
3-2}}
4-3}}
5-4}}
6-5}}
7-7}}
8-8}}
9-9}}
10-11}}

Computation

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.[6]

An alternate method is computing lower and upper bounds with the trapezoidal rule;[7] 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 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

  • Buchstab function, a function used similarly to estimate the number of rough numbers, whose convergence to is controlled by the Dickman function
  • Golomb–Dickman constant

References

1. ^{{cite journal |first=K. |last=Dickman |title=On the frequency of numbers containing prime factors of a certain relative magnitude |journal=Arkiv för Matematik, Astronomi och Fysik |volume=22A |issue=10 |year=1930 |pages=1–14 }}
2. ^{{cite journal |first=N. G. |last=de Bruijn |url=http://alexandria.tue.nl/repository/freearticles/597499.pdf |title=On the number of positive integers ≤ x and free of prime factors > y |journal=Indagationes Mathematicae |volume=13 |year=1951 |pages=50–60 }}
3. ^{{cite journal |first=N. G. |last=de Bruijn |url=http://alexandria.tue.nl/repository/freearticles/597534.pdf |title=On the number of positive integers ≤ x and free of prime factors > y, II |journal=Indagationes Mathematicae |volume=28 |issue= |year=1966 |pages=239–247 }}
4. ^{{cite journal |first=V. |last=Ramaswami |url=http://www.ams.org/bull/1949-55-12/S0002-9904-1949-09337-0/S0002-9904-1949-09337-0.pdf |title=On the number of positive integers less than and free of prime divisors greater than xc |journal=Bulletin of the American Mathematical Society |volume=55 |issue=12 |year=1949 |pages=1122–1127 |doi= 10.1090/s0002-9904-1949-09337-0 | mr=0031958}}
5. ^{{cite journal |first=A. |last=Hildebrand |first2=G. |last2=Tenenbaum |url=http://archive.numdam.org/article/JTNB_1993__5_2_411_0.pdf |title=Integers without large prime factors |journal=Journal de théorie des nombres de Bordeaux |volume=5 |issue=2 |year=1993 |pages=411–484 |doi=10.5802/jtnb.101}}
6. ^{{cite journal |first=Eric |last=Bach |first2=René |last2=Peralta |url=http://cr.yp.to/bib/1996/bach-semismooth.pdf |title=Asymptotic Semismoothness Probabilities |journal=Mathematics of Computation |volume=65 |issue=216 |pages=1701–1715 |year=1996 |doi=10.1090/S0025-5718-96-00775-2 }}
7. ^1 {{cite journal |first=J. |last=van de Lune |first2=E. |last2=Wattel |title=On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory |journal=Mathematics of Computation |volume=23 |issue=106 |year=1969 |pages=417–421 |doi=10.1090/S0025-5718-1969-0247789-3 }}
8. ^{{cite journal |first=George |last=Marsaglia |first2=Arif |last2=Zaman |first3=John C. W. |last3=Marsaglia |title=Numerical Solution of Some Classical Differential-Difference Equations |journal=Mathematics of Computation |volume=53 |issue=187 |year=1989 |pages=191–201 |jstor= |doi=10.1090/S0025-5718-1989-0969490-3 }}
9. ^{{cite journal |first=John B. |last=Friedlander |url=http://plms.oxfordjournals.org/content/s3-33/3/565 |title=Integers free from large and small primes |journal=Proc. London Math. Soc. |volume=33 |issue=3 |pages=565–576 |year=1976 |doi=10.1112/plms/s3-33.3.565 }}

Further reading

  • {{Cite arxiv

|first1=David
|last1=Broadhurst
|title=Dickman polylogarithms and their constants
|eprint=1004.0519
|year=2010
|class=math-ph
}}
  • {{cite journal

|first1=Kannan
|last1=Soundararajan
|title=An asymptotic expansion related to the Dickman function
|arxiv=1005.3494
|year=2012
|journal=Ramanujan Journal
|volume=29
|issue=1–3
|doi=10.1007/s11139-011-9304-3
|mr=2994087
|pages=25–30
}}
  • {{mathworld|urlname=DickmanFunction|title=Dickman function}}
{{DEFAULTSORT:Dickman Function}}

2 : Analytic number theory|Special functions
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