In analytic number theory
, Dickman's function
is a special function
used to estimate the proportion of smooth numbers
up to a given bound.
Dickman's function is named after actuary Karl Dickman, who defined it in his only mathematical publication. Its properties were later studied by the Dutch mathematician Nicolaas Govert de Bruijn;, so some sources call it the Dickman-de Bruijn function.
The Dickman-de Bruijn function
is a continuous function
that satisfies the delay differential equation
with initial conditions
for 0 ≤
≤ 1. Dickman showed heuristically that
is the number of y
integers below x
V. Ramaswami of Andhra University later gave a rigorous proof that was asymptotic to , along with the error bound
in big O notation
The main purpose of the Dickman-de Bruijn function is to estimate the frequency of smooth numbers at a given size. This can be used to optimize various number-theoretical algorithms, and can be useful of its own right.
It can be shown using that
which is related to the estimate
The Golomb–Dickman constant has an alternate definition in terms of the Dickman-de Bruijn function.
A first approximation might be
A better estimate is
where Ei is the exponential integral
and ξ is the positive root of
A simple upper bound is
|| 1 |
|| 3.0685282 |
|| 4.8608388 |
|| 4.9109256 |
|| 3.5472470 |
|| 1.9649696 |
|| 8.7456700 |
|| 3.2320693 |
|| 1.0162483 |
|| 2.7701718 |
For each interval
an integer, there is an analytic function
. For 0 ≤
. For 1 ≤
. For 2 ≤
can be calculated using infinite series.
An alternate method is computing lower and upper bounds with the trapezoidal rule; a mesh of progressively finer sizes allows for arbitrary accuracy. For high precision calculations (hundreds of digits), a recursive series expansion about the midpoints of the intervals is superior.
Bach and Peralta define a two-dimensional analog
. This function is used to estimate a function
similar to de Bruijn's, but counting the number of y
-smooth integers with at most one prime factor greater than z