View source for Dickman-function(2)

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The unique continuous solution of the system

$$ 
\rho ( u ) = 1   ( 0 \leq  u \leq  1 ) ,
$$

$$ 
u \rho  ^  \prime  ( u ) = - \rho ( u - 1 )   ( u > 1 ) .
$$

The Dickman function  $  \rho ( u ) $
occurs in the problem of estimating the number  $  \Psi ( x,y ) $
of positive integers not exceeding  $  x $
that are free of prime factors greater than  $  y $: 
for any fixed  $  u > 0 $, 
one has  $  \Psi ( x,x ^ { {1 / u } } ) \sim \rho ( u ) x $
as  $  u \rightarrow \infty $[[#References|[a2]]], [[#References|[a4]]].

The function  $  \rho ( u ) $
is positive, non-increasing and tends to zero at a rate faster than exponential as  $  u \rightarrow \infty $. 
A precise asymptotic estimate is given by the de Bruijn–Alladi formula [[#References|[a1]]], [[#References|[a3]]]:

$$ 
\rho ( u ) = ( 1 + O ( {
\frac{1}{u}
 } ) ) \sqrt { {
\frac{\xi  ^  \prime  ( u ) }{2 \pi }
 } } \times
$$

$$ 
\times 
{ \mathop{\rm exp} } \left \{ \gamma - u \xi ( u ) + \int\limits _ { 0 } ^ {  \xi   ( u ) } { {
\frac{e  ^ {s} - 1 }{s}
 } }  {ds } \right \}   ( u > 1 ) ,
$$

where  $  \gamma $
is the [[Euler constant|Euler constant]] and  $  \xi ( u ) $
is the unique positive solution of the equation  $  e ^ {\xi ( u ) } = 1 + u \xi ( u ) $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Alladi,   "The Turán–Kubilius inequality for integers without large prime factors"  ''J. Reine Angew. Math.'' , '''335'''  (1982)  pp. 180–196</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.G. de Bruijn,   "On the number of positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110180/d11018017.png" /> and free of prime factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110180/d11018018.png" />"  ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''54'''  (1951)  pp. 50–60</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.G. de Bruijn,   "The asymptotic behaviour of a function occurring in the theory of primes"  ''J. Indian Math. Soc. (N.S.)'' , '''15'''  (1951)  pp. 25–32</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Hildebrand,   G. Tenenbaum,   "Integers without large prime factors"  ''J. de Théorie des Nombres de Bordeaux'' , '''5'''  (1993)  pp. 411–484</TD></TR></table>