Dickman function on arvau

Wikipedia / 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 , who defined it in his only mathematical publication. It was later studied by the Dutch mathematician Nicolaas Govert de Bruijn.

Definition

is a continuous function that satisfies the delay differential equation

for 0≤u≤1.

Properties

is fixed, we have

is the number of y-smooth (or y-friable) integers belowx

Ramaswami later gave a rigorous proof that for fixed a, with the error bound

in big O notation.

Knuth gives a proof for a narrowed bound:

where is Euler's constant.

Applications

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

below.

The Golomb–Dickman constant has an alternate definition in terms of the Dickman–de Bruijn function.

Estimation

A better estimate is

where Ei is the exponential integral and ξ is the positive root of

Computation

For each interval [n−1,n] with n an integer, there is an analytic function. For 0≤u≤. For 1≤u≤. For 2≤u≤3,

with Li2 the dilogarithm 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. Values for u≤7 can be usefully computed via numerical integration in ordinary double-precision floating-point.

Extension

. similar to de Bruijn's, but counting the number of y-smooth integers with at most one prime factor greater than z. Then

This class of numbers may be encountered in the two-stage variant of P-1 factoring. However, Kruppa's estimate of the probability of finding a factor by P-1 does not make use of this result.