In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound. It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, [1] which is not easily available, [2] and later studied by the Dutch mathematician Nicolaas Govert de Bruijn. [3] [4]
The Dickman–de Bruijn function is a continuous function that satisfies the delay differential equation
with initial conditions for 0 ≤ u ≤ 1.
Dickman proved that, when is fixed, we have
where is the number of y- smooth (or y- friable) integers below x.
Ramaswami later gave a rigorous proof that for fixed a, was asymptotic to , with the error bound
in big O notation. [5]
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 such as P–1 factoring and can be useful of its own right.
It can be shown that [6]
which is related to the estimate below.
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 [7]
where Ei is the exponential integral and ξ is the positive root of
A simple upper bound is
1 | 1 |
2 | 3.0685282×10−1 |
3 | 4.8608388×10−2 |
4 | 4.9109256×10−3 |
5 | 3.5472470×10−4 |
6 | 1.9649696×10−5 |
7 | 8.7456700×10−7 |
8 | 3.2320693×10−8 |
9 | 1.0162483×10−9 |
10 | 2.7701718×10−11 |
For each interval [n − 1, n] with n an integer, there is an analytic function such that . For 0 ≤ u ≤ 1, . For 1 ≤ u ≤ 2, . For 2 ≤ u ≤ 3,
with Li2 the dilogarithm. Other can be calculated using infinite series. [8]
An alternate method is computing lower and upper bounds with the trapezoidal rule; [7] 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. [9]
Friedlander defines a two-dimensional analog of . [10] 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. Then
In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound. It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, [1] which is not easily available, [2] and later studied by the Dutch mathematician Nicolaas Govert de Bruijn. [3] [4]
The Dickman–de Bruijn function is a continuous function that satisfies the delay differential equation
with initial conditions for 0 ≤ u ≤ 1.
Dickman proved that, when is fixed, we have
where is the number of y- smooth (or y- friable) integers below x.
Ramaswami later gave a rigorous proof that for fixed a, was asymptotic to , with the error bound
in big O notation. [5]
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 such as P–1 factoring and can be useful of its own right.
It can be shown that [6]
which is related to the estimate below.
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 [7]
where Ei is the exponential integral and ξ is the positive root of
A simple upper bound is
1 | 1 |
2 | 3.0685282×10−1 |
3 | 4.8608388×10−2 |
4 | 4.9109256×10−3 |
5 | 3.5472470×10−4 |
6 | 1.9649696×10−5 |
7 | 8.7456700×10−7 |
8 | 3.2320693×10−8 |
9 | 1.0162483×10−9 |
10 | 2.7701718×10−11 |
For each interval [n − 1, n] with n an integer, there is an analytic function such that . For 0 ≤ u ≤ 1, . For 1 ≤ u ≤ 2, . For 2 ≤ u ≤ 3,
with Li2 the dilogarithm. Other can be calculated using infinite series. [8]
An alternate method is computing lower and upper bounds with the trapezoidal rule; [7] 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. [9]
Friedlander defines a two-dimensional analog of . [10] 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. Then