Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2014.tde-24032015-132813
Document
Author
Full name
Everton Juliano da Silva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2014
Supervisor
Committee
Martin, Paulo Agozzini (President)
Bertato, Fabio Maia
Simonis, Adilson
Title in Portuguese
Uma demonstração analÃtica do teorema de Erdös-Kac
Keywords in Portuguese
Desigualdade de Berry-Esseen
Método de Selberg-Delange
Teorema da continuidade de Lévy
Teorema de Erdös-Kac
Teoria probabilÃstica dos números
Abstract in Portuguese
Em teoria dos números, o teorema de Erdös-Kac, também conhecido como o teorema fundamental de teoria probabilÃstica dos números, diz que se w(n) denota a quantidade de fatores primos distintos de n, então a sequência de funções de distribuições N definidas por FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converge uniformemente sobre R para a distribuição normal padrão. Neste trabalho desenvolvemos todos os teoremas necessários para uma demonstração analÃtica, que nos permitirá encontrar a ordem de erro da convergência acima.
Title in English
An analytic proof of Erdös-Kac theorem
Keywords in English
Berry-Esseen inequality
Erdös-Kac theorem
Lévy´s continuity theorem
Probabilistic number theory
Selberg-Delange method
Abstract in English
In number theory, the Erdös-Kac theorem, also known as the fundamental theorem of probabilistic number theory, states that if w(n) is the number of distinct prime factors of n, then the sequence of distribution functions N, defined by FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converges uniformly on R to the standard normal distribution. In this work we developed all theorems needed to an analytic demonstration, which will allow us to find an order of error of the above convergence.
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Publishing Date
2015-06-03