Erdős–Kac theorem

The set of Erdős - Kac [ ɛrdø ː ʃ - kaʦ ] by Paul Erdős and Mark Kac is a set of number theory and states that the number of different prime factors of a randomly drawn number from the set for large is approximately normally distributed. The same result holds for the counted with multiplicity prime factors.

More precisely, if the total number of different prime factors of the number designated for fixed with

In which

The probability density function of the standard normal distribution, which often occurs as a limit value of distribution of probability and statistics.

Heuristic motivation

Are and two different prime numbers and is a large number, so each is composed of numbers from 1 to equally probable number drawn with probability approximately by, about and with probability by approximately divisible by and with probability. The events and are therefore nearly stochastically independent. The function can be approximately as the sum of independent indicator functions

Represent and should therefore be approximated by the normal distribution for large.

History

The theorem is a generalization of the theorem of Hardy - Ramanujan on the average asymptotic number of prime factors. Erdős heard the sentence as Kac conjecture in a lecture at Princeton pronounce and came shortly after the end of the lecture with the evidence. The set was released in 1940 by Erdős and Kac, remained largely unnoticed for ten years and in 1958 was proved by Alfréd Rényi and Paul Turán in a version with an explicit error term. According to Kac marks the sentence " the collection of the law of normal distribution [ ... ] in the theory of numbers and was the birth of a new branch of this ancient discipline," the probabilistic number theory.