Dickman function
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
is the number of y-smooth (or y-friable) integers below x. Ramaswami later gave a rigorous proof that for fixed a,
, 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
The Golomb–Dickman constant has an alternate definition in terms of the Dickman–de Bruijn function.
A better estimate is[7] where Ei is the exponential integral and ξ is the positive root of A simple upper bound is
For each interval [n − 1, n] with n an integer, there is an analytic function
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
similar to de Bruijn's, but counting the number of y-smooth integers with at most one prime factor greater than z.
