Dickman-function(2)
From Encyclopedia of Mathematics
The unique continuous solution of the system
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 |
The Dickman function 
occurs in the problem of estimating the number 
of positive integers not exceeding 
that are free of prime factors greater than 
: for any fixed 
, one has 
as 
[a2], [a4].
The function 
is positive, non-increasing and tends to zero at a rate faster than exponential as 
. A precise asymptotic estimate is given by the de Bruijn–Alladi formula [a1], [a3]:
 |
 |
where 
is the Euler constant and 
is the unique positive solution of the equation 
.
References
| [a1] | K. Alladi, "The Turán–Kubilius inequality for integers without large prime factors" J. Reine Angew. Math. , 335 (1982) pp. 180–196 |
| [a2] | N.G. de Bruijn, "On the number of positive integers  and free of prime factors  " Nederl. Akad. Wetensch. Proc. Ser. A , 54 (1951) pp. 50–60 |
| [a3] | 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 |
| [a4] | A. Hildebrand, G. Tenenbaum, "Integers without large prime factors" J. de Théorie des Nombres de Bordeaux , 5 (1993) pp. 411–484 |
How to Cite This Entry:
Dickman-function(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickman-function(2)&oldid=46649
Dickman-function(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickman-function(2)&oldid=46649
This article was adapted from an original article by A. Hildebrand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article