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| [[Image:DickmanRho.png|thumb|right|400px|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 [[Linearithmic function|quasilinear]].]]
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| In [[analytic number theory]], the '''Dickman function''' or '''Dickman–de Bruijn function''' ρ is a [[special function]] used to estimate the proportion of [[smooth number]]s up to a given bound.
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| It was first studied by actuary [[Karl Dickman]], who defined it in his only mathematical publication,<ref>{{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 }}</ref> and later studied by the Dutch mathematician [[Nicolaas Govert de Bruijn]].<ref>{{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 }}</ref><ref>{{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 }}</ref>
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| ==Definition==
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| The Dickman-de Bruijn function <math>\rho(u)</math> is a [[continuous function]] that satisfies the [[delay differential equation]]
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| :<math>u\rho'(u) + \rho(u-1) = 0\,</math>
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| with initial conditions <math>\rho(u) = 1</math> for 0 ≤ ''u'' ≤ 1. Dickman proved that, when <math> a </math> is fixed, we have
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| :<math>\Psi(x, x^{1/a})\sim x\rho(a)\,</math> | |
| where <math>\Psi(x,y)</math> is the number of ''y''-[[Smooth number|smooth]] (or ''y''-[[Friable number|friable]]) integers below ''x''.
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| V. Ramaswami of [[Andhra University]] later gave a rigorous proof that <math>\Psi(x,x^{1/a})</math> was asymptotic to <math>x \rho(a)</math>, with the [[error bound]]
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| :<math>\Psi(x,x^{1/a})=x\rho(a)+O(x/\log x)</math> | |
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| in [[big O notation]].<ref>{{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 <math>x</math> and free of prime divisors greater than ''x''<sup>''c''</sup> |journal=Bulletin of the American Mathematical Society |volume=55 |issue= |year=1949 |pages=1122–1127 |doi= }}</ref>
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| ==Applications==
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| 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.
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| It can be shown using <math>\log\rho</math> that<ref>{{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 }}</ref>
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| :<math>\Psi(x,y)=xu^{O(-u)}</math>
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| which is related to the estimate <math>\rho(u)\approx u^{-u}</math> below.
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| The [[Golomb–Dickman constant]] has an alternate definition in terms of the Dickman–de Bruijn function.
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| ==Estimation==
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| A first approximation might be <math>\rho(u)\approx u^{-u}.\,</math> A better estimate is<ref name="vandeLuneWattel" />
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| :<math>\rho(u)\sim\frac{1}{\xi\sqrt{2\pi u}}\cdot\exp(-u\xi+\operatorname{Ei}(\xi))</math>
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| where Ei is the [[exponential integral]] and ξ is the positive root of
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| :<math>e^\xi-1=u\xi.\,</math>
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| A simple upper bound is <math>\rho(x)\le1/x!.</math>
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| {| class="wikitable" style="float:right"
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| |-
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| ! <math>u</math>
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| ! <math>\rho(u)</math>
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| |-
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| | 1
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| | 1
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| |-
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| | 2
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| | 3.0685282{{e|-1}}
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| |-
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| | 3
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| | 4.8608388{{e|-2}}
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| |-
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| | 4
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| | 4.9109256{{e|-3}}
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| |-
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| | 5
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| | 3.5472470{{e|-4}}
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| |-
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| | 6
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| | 1.9649696{{e|-5}}
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| |-
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| | 7
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| | 8.7456700{{e|-7}}
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| |-
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| | 8
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| | 3.2320693{{e|-8}}
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| |-
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| | 9
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| | 1.0162483{{e|-9}}
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| |-
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| | 10
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| | 2.7701718{{e|-11}}
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| |}
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| ==Computation==
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| For each interval [''n'' − 1, ''n''] with ''n'' an integer, there is an analytic function <math>\rho_n</math> such that <math>\rho_n(u)=\rho(u)</math>. For 0 ≤ ''u'' ≤ 1, <math>\rho(u) = 1</math>. For 1 ≤ ''u'' ≤ 2, <math>\rho(u) = 1-\log u</math>. For 2 ≤ ''u'' ≤ 3,
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| :<math>\rho(u) = 1-(1-\log(u-1))\log(u) + \operatorname{Li}_2(1 - u) + \frac{\pi^2}{12}</math>.
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| with Li<sub>2</sub> the [[Polylogarithm#Dilogarithm|dilogarithm]]. Other <math>\rho_n</math> can be calculated using infinite series.<ref name="BachPeralta">{{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 }}</ref>
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| An alternate method is computing lower and upper bounds with the [[trapezoidal rule]];<ref name="vandeLuneWattel">{{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 }}</ref> 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.<ref>{{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 }}</ref>
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| ==Extension==
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| Bach and Peralta define a two-dimensional analog <math>\sigma(u,v)</math> of <math>\rho(u)</math>.<ref name="BachPeralta" /> This function is used to estimate a function <math>\Psi(x,y,z)</math> similar to de Bruijn's, but counting the number of ''y''-smooth integers with at most one prime factor greater than ''z''. Then
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| :<math>\Psi(x,x^{1/a},x^{1/b})\sim x\sigma(b,a).\,</math> | |
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| ==References== | |
| <references/>
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| ==External links==
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| * {{Cite arxiv
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| |first1=David
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| |last1=Broadhurst
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| |title=Dickman polylogarithms and their constants
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| |eprint=1004.0519
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| |year=2010
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| }}
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| * {{ Cite arxiv
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| |first1=K.
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| |last1=Soundararajan
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| |title=An asymptotic expansion related to the Dickman function
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| |eprint=1005.3494
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| |year=2010
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| }}
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| * {{mathworld|urlname=DickmanFunction|title=Dickman function}}
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| {{DEFAULTSORT:Dickman Function}}
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| [[Category:Analytic number theory]]
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| [[Category:Special functions]]
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