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]:
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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 ![]() ![]() |
[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=15579
Dickman-function(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickman-function(2)&oldid=15579
This article was adapted from an original article by A. Hildebrand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article