# Dickman-function(2)

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

The unique continuous solution of the system

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

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

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 $u \rightarrow \infty$[a2], [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 [a1], [a3]:

$$\rho ( u ) = ( 1 + O ( { \frac{1}{u} } ) ) \sqrt { { \frac{\xi ^ \prime ( u ) }{2 \pi } } } \times { \mathop{\rm exp} } \left \{ \gamma - u \xi ( u ) + \int\limits _ { 0 } ^ { \xi ( u ) } { { \frac{e ^ {s} - 1 }{s} } } {ds } \right \} \quad ( u > 1 ) ,$$

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

#### 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 $\leq x$ and free of prime factors $> y$" 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=54294
This article was adapted from an original article by A. Hildebrand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article