View source for Dickman function

The function $\rho(u)$ defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300902.png" /> by the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300903.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300904.png" /> and by the [[differential-delay equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300905.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300906.png" />. Interest attaches to this function because of its connection to  "smooth"  numbers, i.e. numbers that are the product of many small prime numbers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300907.png" /> denote the number of positive integers less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300908.png" /> and free of prime divisors greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d1300909.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009010.png" /> is much larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009011.png" />, it is a simple matter of inclusion-and-exclusion counting (cf. also [[Inclusion-exclusion formula|Inclusion-exclusion formula]]) to show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009013.png" /> denotes a [[Prime number|prime number]]. But the error terms grow rapidly, and the  "main"  term gives the wrong answer in the ranges of greatest interest, including the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009014.png" /> is comparable to a fixed fractional power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009015.png" />. For this case, K. Dickman found that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009016.png" />. If, in place of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009018.png" /> one takes the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009019.png" />, the resulting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009020.png" /> is approximated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009022.png" /> is the [[Buchstab function]], defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009026.png" />, where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009027.png" /> the right-hand derivative has to be taken, [[#References|[a1]]]. Unlike <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009029.png" /> oscillates and tends to a positive limit, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009030.png" />.

There are two combinatorial identities linking the Dickman function to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009031.png" />. Early work is based on the [[Buchstab identity]]: With <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009032.png" /> denoting a prime number, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009033.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009034.png" /></td> </tr></table>

The usual heuristic device of replacing a sum over prime numbers by an integral with  "prime density"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009035.png" /> and replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009036.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009037.png" /> leads to an identity which, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009039.png" />, simplifies to an integral equivalent to the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009040.png" />. N.G. de Bruijn carried this idea to its limits in [[#References|[a2]]], but accuracy suffers when large and comparable estimated quantities must be subtracted. The more recent Hildebrand identity involves only additions and has the further advantage that the second input is the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009041.png" /> throughout:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009042.png" /></td> </tr></table>

Applications require estimates uniform in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009043.png" />; the best known estimate along these lines, due to A. Hildebrand and based on the identity above, is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009044.png" /></td> </tr></table>

uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009046.png" />. There are similar results for algebraic integers, [[#References|[a3]]].

There are also results concerning the number of smooth integers in an interval, and concerning the distribution of smooth integers into congruence classes [[#References|[a4]]], [[#References|[a5]]].

The Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]) implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009047.png" /> [[#References|[a8]]].

The analytical properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009048.png" /> are reasonably well understood; calculus, analysis of the Laplace transform, and the saddle-point method are the key tools. The first extensive analysis is due to De Bruijn, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009049.png" /> is sometimes termed the Dickman–De Bruijn function. One has [[#References|[a10]]]:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009050.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009051.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009052.png" /> is positive for all positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009053.png" />, and hence, from the definition, decreasing);

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009054.png" /> is log-concave, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009057.png" /> (K. Alladi);

c) The [[Laplace transform|Laplace transform]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009058.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009059.png" /></td> </tr></table>

d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009060.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009061.png" />.

The Dickman function is one of a parameterized family of related functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009062.png" />, [[#References|[a12]]], and a wider class of similar delay-differential equations has been studied in [[#References|[a7]]]. A quick and simple bit of Mathematica code suffices to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009063.png" /> to reasonable accuracy in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009064.png" /> using step-size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009065.png" />. This code calculates a table of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009066.png" /> at intervals of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009067.png" />, working back recursively into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009068.png" />:

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009069.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009070.png" />,</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009071.png" />;</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009072.png" />;</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009073.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130090/d13009074.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.G. de Bruijn,   "On the number of positive integers $\leq x$ and free prime factors $> y$. II"  ''Indag. Math.'' , '''28'''  (1966)  pp. 239–247  ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''69'''  (1966)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  N.G. de Bruijn,   "On the number of uncancelled elements in the sieve of Eratosthenes"  ''Indag. Math.'' , '''12'''  (1950)  pp. 247–256  ''Nederl. Akad. Wetensch. Proc.'' , '''53'''  (1950)  pp. 803–812</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Friedlander,   "On the number of ideals free from large prime divisors"  ''J. Reine Angew. Math.'' , '''255'''  (1972)  pp. 1–7</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Friedlander,   A. Granville,   "Integers without large prime factors, in short intervals"  ''Philos. Trans. Royal Soc.'' , '''345'''  (1993)  pp. 339–348</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Granville,   "On integers, without large prime factors, in arithmetic progressions II"  ''Philos. Trans. Royal Soc.'' , '''345'''  (1993)  pp. 349–362</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Hildebrand,   G. Tenenbaum,   "Integers without large prime factors"  ''J. Théor. Nombres Bordeaux'' , '''5''' :  2  (1993)  pp. 411–484</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Hildebrand,   G. Tenenbaum,   "On a class of differential-difference equations arising in number theory"  ''J. Anal. Math.'' , '''61'''  (1993)  pp. 145–179</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Hildebrand,   "Integers free of large prime factors and the Riemann hypothesis"  ''Mathematika'' , '''31''' :  2  (1985)  pp. 258–271</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  S. Hunter,   J. Sorenson,   "Approximating the number of integers free of large prime factors"  ''Math. Comput.'' , '''66''' :  220  (1997)  pp. 1729–1741</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  P. Moree,   "Psixyology and Diophantine equations"  ''Diss. Univ. Leiden''  (1993)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  E. Saias,   "Sur le nombre des entiers sans grand facteur premier"  ''J. Number Theory'' , '''32''' :  1  (1989)  pp. 78–99</TD></TR>
<TR><TD valign="top">[a12]</TD> <TD valign="top">  F.S. Wheeler,   "Two differential-difference equations arising in number theory"  ''Trans. Amer. Math. Soc.'' , '''318''' :  2  (1990)  pp. 491–523</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  D.S. Mitrinović,   J. Sandor,   B. Crstici,   "Handbook of number theory" , Kluwer Acad. Publ.  (1996)  pp. Sect. IV.21</TD></TR>
<TR><TD valign="top">[a14]</TD> <TD valign="top">  K. Dickman,   "On the frequency of numbers containing prime factors of a certain relative magnitude"  ''Ark. Mat., Astron. och Fysik'' , '''22A''' :  10  (1930)  pp. 1–14</TD></TR></table>

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