Andrew Granville, K. Soundararajan
We give a relatively easy proof of the Erd\H os-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.
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The Erdős-Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this paper, we establish the large deviations and moderate deviations for this problem in a very general setting for a wide class of additive functions.
We study probability distributions arising from local obstructions to the existence of $p$-adic points in families of varieties. In certain cases we show that an Erdős-Kac type law holds.
Erdős-Kac theorems for discriminants of number fields
本文证明了当 G 为阿贝尔群时，随机 G-扩张中分歧素数个数的中心极限定理，推广了 Lemke Oliver 和 Thorne 在 G=S_d (2≤d≤5) 情形的工作，并首次给出不同素数处分歧事件不独立的例子。
High and odd moments in the Erdős--Kac theorem
本文证明Erdős--Kac定理中Granville和Soundararajan给出的矩估计范围是精确的，并首次给出奇数阶矩在k=O((log log x)^{1/3})时的渐近公式，还发现泊松分布的中心矩可极好地近似Erdős--Kac矩。
We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erdös-Kac theorem for additive arithmetic functions $ψ$ of a random integer $J_n$ uniformly distributed over $\{1,...,n\}$. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods, and our $ψ$ is more general than those considered in the literature. Our main results are (i) bounds on the Kolmogorov distance and Wasserstein distance between the distribution of the normalized $ψ(J_n)$ and the standard Gaussian distri…
Let $x\geqslant 3$. For $1\leqslant n\leqslant x$ an integer, let $ω(n)$ be its number of distinct prime factors. We show that $ω(n-1)$ satisfies an Erdős-Kac type theorem whenever $ω(n)=k$ where $1\leqslant k\ll\log\log x$, thus extending a result of Halberstam.
For positive integers $k$ and $n$ let $σ_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number $α_k = \sum_{n\geq 1} \frac{σ_k(n)}{n!}$ is irrational. It is known unconditionally that $α_k$ is irrational if $k\leq 3$. We prove $α_4$ is irrational.
On an Erdős--Kac-type conjecture of Elliott
本文证明了Elliott基于Erdős--Kac定理提出的一个关于素数求和渐近性的猜想可由Bombieri--Vinogradov定理推出，并利用素数更深层的分布水平结果证明了一个涉及Poisson--Dirichlet分布的相关结论。
Generalizations of Erdős-Kac theorem with applications
本文将 Erdős-Kac 定理从阿贝尔幺半群推广到满足附加条件的任意子集，统一研究 omega 函数与除数函数等算术函数，并应用于数域、整体函数域和几何不可约射影簇。
We prove an Erdős-Kac type of theorem for the set $S(x,y)=\{n\leq x: p|n \Rightarrow p\leq y \}$. If $ω(n)$ is the number of prime factors of $n$, we prove that the distribution of $ω(n)$ for $n \in S(x,y)$ is Gaussian for a certain range of $y$ using method of moments. The advantage of the present approach is that it recovers classical results for the range $u=o(\log \log x )$ where $u=\frac{\log x}{\log y}$, with a much simpler proof.
We show that the Erdős-Kac theorem is valid in almost all intervals $\left[x,x+h\right]$ as soon as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log\log x$, almost all interval $\left[x,x+\exp\left(\left(\log\log x\right)^{1/2+\varepsilon}\right)\right]$ contains the expected number of integers $n$ such that $ω(n)=k$. These results are a consequence of the methods introduced by Matomäki and Radziwiłł to estimate sums of multiplicative functions over short intervals.
Let $s(n)=\sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. The second-named author proved that $ω(s(n))$ has normal order $\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erdős--Kac theorem: $ω(s(n))$ is asymptotically normally distributed with mean and variance $\log\log{n}$. The same method applies with $s(n)$ replaced by any of several other unconventional arithmetic functions, such as $β(n):=\sum_{p\mid n} p$, $n-ϕ(n)$, and $n+τ(n)$ ($τ$ being the divisor function).
We investigate the number of prime factors of individual entries for matrices in the special linear group over the integers. We show that, when properly normalised, it satisfies a central limit theorem of Erdős-Kac-type. To do so, we employ a sieve-theoretic set-up due to Granville and Soundararajan. We also make use of an estimate coming from homogeneous dynamics due to Gorodnik and Nevo.
A subset generalization of the Erdős-Kac theorem over number fields with applications
本文将 Erdős-Kac 定理从阿贝尔幺半群推广到数域理想集合中满足附加条件的任意子集，并据此证明了数域上 h-free 与 h-full 理想的 Erdős-Kac 定理。
Erdős-Kac type theorem for the number of scattering geodesics on modular surface
本文受经典 Erdős-Kac 定理启发，证明模曲面上具有共同停留时间的散射测地线数量满足一种 Erdős-Kac 型高斯分布规律。
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 …
An Erdős-Kac theorem for integers with dense divisors
本文证明了连续因子之比有上界的大整数其素因子个数服从近似正态分布，并将该结果在两个不同方向上进行了推广。
Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
本文研究整数素数因子分布与最大素因子分布的线性无关性，改进了埃罗什-卡茨定理和洛伊德的结果，并展示了这些结果在埃罗什-波默兰茨定理下的类比性。
Transforming the Erdős-Kac theorem
本文通过 delta 方法变换 Erdős-Kac 定理，构造方差稳定、宽度最优的 score 等区间估计，并用模糊覆盖概率验证其在小正整数上的可靠性。
Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers
该论文证明了在加权$2^{\omega(n-1)}$下，具有$k$个素因子的整数平移$n-1$的素因子个数满足大偏差区间的Erdős-Kac定理，并给出拟最优误差项，改进了已有结果。
Let $2 \leq y \leq x$ such that $β:= \frac{\log x}{\log y} \rightarrow \infty$. Let $ω_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p \leq y$, and let $μ_y(n) := μ^2(n)(-1)^{ω_y(n)}$, where $μ$ is the Möbius function. We prove that if $β$ is not too large (in terms of $x$) then for each fixed $a \in \mathbb{N}$, \begin{equation*} \sum_{n \leq x} μ_y(n)μ_y(n+a) \ll x\left(\frac{1}{\log_2 y} + e^{-\frac{1}{21}β\log β}\right). \end{equation*} This can be seen as a partial result towards the binary Chowla conjecture. Our main input is a \emph{quantitative} bivariate anal…
Weighted Erdős-Kac Theorems via Computing Moments
本文推广了计算矩的方法，研究了一大类非负乘性函数加权的加性函数的分布，建立了矩的一致渐近公式，并以此推广了Delange-Halberstam定理，获得了Kubilius-Shapiro定理的加权类似物。
Fix a number field $K$. For each nonzero $α\in \mathbb{Z}_K$, let $ν(α)$ denote the number of distinct, nonassociate irreducible divisors of $α$. We show that $ν(α)$ is normally distributed with mean proportional to $(\log\log |N(α)|)^{D}$ and standard deviation proportional to $(\log\log{|N(α)|})^{D-1/2}$. Here $D$, as well as the constants of proportionality, depend only on the class group of $K$. For example, for each fixed real $λ$, the proportion of $α\in \mathbb{Z}[\sqrt{-5}]$ with $$ ν(α) \le \frac{1}{8}(\log\log{N(α)})^2 + \fracλ{2\sqrt{2}} (\log\log{N(α)})^{3/2} $$ is given by $\frac{…
Multivariate and quantitative Erdős-Kac laws for Beatty sequences
本文研究了Beatty序列的多变量和定量Erdős-Kac定律，证明了其联合分布收敛于高斯分布，并给出了收敛速率的定量界。
A Generalization of the Erdős-Kac Theorem
本文通过扰动均匀概率质量并给出误差项充分条件，将Erdős-Kac定理推广至更一般分布，并应用于Harmonic(n)与Zeta(s)分布。
Some ergodic theorems over $k$-full numbers
本文在先前关于无平方数的遍历定理基础上，将Bergelson-Richter定理、Erdős-Kac定理和Loyd定理推广到$k$-满数情形，对任意整数$k\geq2$成立。
The celebrated Erdős--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to hold for certain non-additive functions and for amenable arithmetic functions over certain subsets of the natural numbers. Continuing in this vein, we show that if $g_1(n), \ldots, g_k(n)$ is a collection of functions satisfying certain mild hypotheses for which an Erdős--Kac-type normal distribution law holds, and if $Q(x_1, \ldots, x_k)$ is a polynomial with no…
Chowla Conjecture for the truncated Möbius function over $\mathbb{F}_q[t]$
本文在函数域固定$q$情形下，通过Kubilius模型和纯Brun筛法，以特征函数形式发展均匀二元Erdős-Kac定理，解析证明了截断Möbius函数的两点Chowla猜想。
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