论文标题
ERDőS-KAC定理的概括
A Generalization of the Erdős-Kac Theorem
论文作者
论文摘要
给定$ n \ in \ mathbb {n} $，令$ω\ left（n \ right）$表示$ n $的不同主要因素的数量，令$ z $表示标准的常规变量，让$ p_ {n} $表示$ \ weft \ weft \ weft \ weft \ weft \ ldots \ ldots，\ ldots n \ ldots n \ n \ right \ right \ rigr。 erdős-kac定理指出，$$ p_ {n} \ left（m \ le n：ω\ left（m \ firt） - \ log \ log \ log \ log n \ le x \ left（\ log \ log \ log \ log n \ right）^{1/2}} {1/2} \ right） $ n \ to \ infty $;即，如果$ n \ left（n \右）$是$ \ lbrace 1，\ ldots，n \ rbrace $，然后$ω\ left（n \ left（n \ orirt）\ oright）$是$ n \ n同于$ n的$ for n y nogrianciance的$ hemptotical n is均匀分布（n \ left（n \ first）\ form y for $ \ for $）本文的贡献是通过考虑以下意义，将ERDőS-KAC定理的概括对较大类的随机变量进行概括对较大类的随机变量。用$ \ mathbb {p} _ {n} $ $ \ left \ {1，\ ldots，n \ right \} $在$ \ weft \ {1，n \ right \} $上表示概率分布通过在$ \ varepsilon_ {i，n} $的情况下提供一些约束，说明了足够的条件，以便得出结论，$$ \ mathbb {p} _ {n} _ {n} \ left（m \ le n：ω\ left（m \ left（m \ right） - n \ right）^{1/2} \ right）\ to \ mathbb {p} \ left（z \ le x \ right）$ $ as $ n \ to \ infty。 主要结果将应用于证明正整数的不同主要因素与谐波$ \ left（n \ firt）$分布或zipf $ \ left（n，s \ firt）$分布也趋向于正常分布$ \ nathcal {n} n} \ loft（\ log n，\ log n，\ log n，\ log n，\ right as \ right as as us）在ZIPF变量的情况下，$ s \ to1 $）。
Given $n\in\mathbb{N}$, let $ω\left(n\right)$ denote the number of distinct prime factors of $n$, let $Z$ denote a standard normal variable, and let $P_{n}$ denote the uniform distribution on $\left\{ 1,\ldots,n\right\} $. The Erdős-Kac Theorem states that $$P_{n}\left(m\le n:ω\left(m\right)-\log\log n\le x\left(\log\log n\right)^{1/2}\right)\to\mathbb{P}\left(Z\le x\right)$$ as $n\to\infty$; i.e., if $N\left(n\right)$ is a uniformly distributed variable on $\lbrace 1,\ldots,n \rbrace$, then $ω\left(N\left(n\right)\right)$ is asymptotically normally distributed as $n\to \infty$ with both mean and variance equal to $\log \log n$. The contribution of this paper is a generalization of the Erdős-Kac Theorem to a larger class of random variables by considering perturbations of the uniform probability mass $\frac{1}{n}$ in the following sense. Denote by $\mathbb{P}_{n}$ a probability distribution on $\left\{ 1,\ldots,n\right\} $ given by $\mathbb{P}_{n}\left(i\right)=\frac{1}{n}+\varepsilon_{i,n}$. By providing some constraints on the $\varepsilon_{i,n}$'s, sufficient conditions are stated in order to conclude that $$\mathbb{P}_{n}\left(m\le n:ω\left(m\right)-\log\log n\le x\left(\log\log n\right)^{1/2}\right) \to \mathbb{P}\left(Z\le x\right)$$ as $n\to\infty.$ The main result will be applied to prove that the number of distinct prime factors of a positive integer with either the Harmonic$\left(n\right)$ distribution or the Zipf$\left(n,s\right)$ distribution also tends to the normal distribution $\mathcal{N}\left(\log\log n,\log\log n\right)$ as $n\to\infty$ (and as $s\to1$ in the case of a Zipf variable).
