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The unique continuous solution of the system

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 .


[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:
This article was adapted from an original article by A. Hildebrand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article