Dickman-de Bruijn function - Revision history

Vipul: /* Definition */

2010-02-09T02:40:46Z

Definition

← Older revision		Revision as of 02:40, 9 February 2010
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	* <math>\rho(u) = 1 - \log u</math> for <math>u \in [1,2]</math>.		* <math>\rho(u) = 1 - \log u</math> for <math>u \in [1,2]</math>.
	* <math>\rho</math> is (strictly) decreasing for <math>u \in [1,\infty)</math>, i.e., <math>\rho'(u) < 0</math> for <math>u \in (1, \infty)</math>.		* <math>\rho</math> is (strictly) decreasing for <math>u \in [1,\infty)</math>, i.e., <math>\rho'(u) < 0</math> for <math>u \in (1, \infty)</math>.
	* <math>\rho</math> is once differentiable on <math>(1,\infty)</math>. More generally, $\rho</math> is <math>n</math> times differentiable everywhere except at the points <math>\{ 1,2,\dots, n \}</math>.		* <math>\rho</math> is once differentiable on <math>(1,\infty)</math>. More generally, <math>\rho</math> is <math>n</math> times differentiable everywhere except at the points <math>\{ 1,2,\dots, n \}</math>.
	* <math>\rho</math> is infinitely differentiable except at integers.		* <math>\rho</math> is infinitely differentiable except at integers.
	* <math>\lim_{u \to \infty} \rho(u) = 0</math>.		* <math>\lim_{u \to \infty} \rho(u) = 0</math>.

	It turns out that the density of numbers with no prime divisor greater than the <math>x^{th}</math> root is given by <math>\rho(x)</math>. Formally, consider, for any <math>N</math>, the fraction of natural numbers <math>n \le N</math> such that all prime divisors of <math>n</math> are at most <math>N^{1/x}</math>. Then, as <math>N \to \infty</math>, this fraction tends to <math>\rho(x)</math>. Thus, this function is crucial to understand the behavior of the [[largest prime divisor]] function and it is important in obtaining [[smooth number\|smoothness bounds]].		It turns out that the density of numbers with no prime divisor greater than the <math>x^{th}</math> root is given by <math>\rho(x)</math>. Formally, consider, for any <math>N</math>, the fraction of natural numbers <math>n \le N</math> such that all prime divisors of <math>n</math> are at most <math>N^{1/x}</math>. Then, as <math>N \to \infty</math>, this fraction tends to <math>\rho(x)</math>. Thus, this function is crucial to understand the behavior of the [[largest prime divisor]] function and it is important in obtaining [[smooth number\|smoothness bounds]].

Vipul at 02:39, 9 February 2010

2010-02-09T02:39:51Z

← Older revision		Revision as of 02:39, 9 February 2010
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			==History==

			The function was first introduced by Dickman with a heuristic argument relating it to smoothness. de Bruijn explored many properties of this function, and Ramaswami gave a formal proof of its relation to the size of the largest prime divisor.

	==Definition==		==Definition==

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	* <math>\rho</math> is once differentiable on <math>(1,\infty)</math>. More generally, $\rho</math> is <math>n</math> times differentiable everywhere except at the points <math>\{ 1,2,\dots, n \}</math>.		* <math>\rho</math> is once differentiable on <math>(1,\infty)</math>. More generally, $\rho</math> is <math>n</math> times differentiable everywhere except at the points <math>\{ 1,2,\dots, n \}</math>.
	* <math>\rho</math> is infinitely differentiable except at integers.		* <math>\rho</math> is infinitely differentiable except at integers.
	* <math>\lim_{u \to \infty) \rho(u) = 0</math>.		* <math>\lim_{u \to \infty} \rho(u) = 0</math>.

	~~==Related facts==~~

	It turns out that the density of numbers with no prime divisor greater than the <math>x^{th}</math> root is given by <math>\rho(x)</math>. Formally, consider, for any <math>N</math>, the fraction of natural numbers <math>n \le N</math> such that all prime divisors of <math>n</math> are at most <math>N^{1/x}</math>. Then, as <math>N \to \infty</math>, this fraction tends to <math>\rho(x)</math>.		It turns out that the density of numbers with no prime divisor greater than the <math>x^{th}</math> root is given by <math>\rho(x)</math>. Formally, consider, for any <math>N</math>, the fraction of natural numbers <math>n \le N</math> such that all prime divisors of <math>n</math> are at most <math>N^{1/x}</math>. Then, as <math>N \to \infty</math>, this fraction tends to <math>\rho(x)</math>. Thus, this function is crucial to understand the behavior of the [[largest prime divisor]] function and it is important in obtaining [[smooth number\|smoothness bounds]].

Vipul: Created page with '==Definition== This function, called '''Dickman's function''' or the '''Dickman-de Bruijn function''', is defined as the function $\rho:[0,\infty) \to \R$ satisfying …'

2010-02-09T02:37:53Z

Created page with '==Definition== This function, called '''Dickman's function''' or the '''Dickman-de Bruijn function''', is defined as the function <math>\rho:[0,\infty) \to \R</math> satisfying …'

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==Definition==

This function, called '''Dickman's function''' or the '''Dickman-de Bruijn function''', is defined as the function <math>\rho:[0,\infty) \to \R</math> satisfying the delay differential equation:

<math>\! u\rho'(u) + \rho(u - 1) = 0</math>

subject to the initial condition <math>\rho(u) = 1</math> for <math>u \in [0,1]</math>. The function satisfies the following properties:

* <math>\rho(u) = 1</math> for <math>u \in [0,1]</math>.
* <math>\rho(u) = 1 - \log u</math> for <math>u \in [1,2]</math>.
* <math>\rho</math> is (strictly) decreasing for <math>u \in [1,\infty)</math>, i.e., <math>\rho'(u) < 0</math> for <math>u \in (1, \infty)</math>.
* <math>\rho</math> is once differentiable on <math>(1,\infty)</math>. More generally, $\rho</math> is <math>n</math> times differentiable everywhere except at the points <math>\{ 1,2,\dots, n \}</math>.
* <math>\rho</math> is infinitely differentiable except at integers.
* <math>\lim_{u \to \infty) \rho(u) = 0</math>.

==Related facts==

It turns out that the density of numbers with no prime divisor greater than the <math>x^{th}</math> root is given by <math>\rho(x)</math>. Formally, consider, for any <math>N</math>, the fraction of natural numbers <math>n \le N</math> such that all prime divisors of <math>n</math> are at most <math>N^{1/x}</math>. Then, as <math>N \to \infty</math>, this fraction tends to <math>\rho(x)</math>.

Dickman-de Bruijn function - Revision history

Vipul: /* Definition */

Vipul at 02:39, 9 February 2010

Vipul: Created page with '==Definition== This function, called '''Dickman's function''' or the '''Dickman-de Bruijn function''', is defined as the function \rho:[0,\infty) \to \R satisfying …'

Vipul: Created page with '==Definition== This function, called '''Dickman's function''' or the '''Dickman-de Bruijn function''', is defined as the function $\rho:[0,\infty) \to \R$ satisfying …'