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).
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.
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.
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.
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.
Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Ohio State University（俄亥俄州立大学）；University of Arizona（亚利桑那大学）；Xiamen University（厦门大学）
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…
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{…
Let $m\geq 2$ be a positive integer. Given a set $E(ω)\subseteq \mathbb{N}$ we define $r_{N}^{(m)}(ω)$ to be the number of ways to represent $N\in \mathbb{Z}$ as any combination of sums $\textit{ and }$ differences of $m$ distinct elements of $E(ω)$. In this paper, we prove the existence of a "thick" set $E(ω)$ and a positive constant $K$ such that $r_{N}^{(m)}(ω)<K$ for all $N\in \mathbb{Z}$. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
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…
A set of integers greater than 1 is primitive if no element divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erdős question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements.
In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for S_d-extensions that the number of such primes is normally distributed with mean and variance log log X.
Erd\H os and Rényi claimed and Vu proved that for all $h \ge 2$ and for all $ε> 0$, there exists $g = g_h(ε)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$ elements of $A$ is bounded by $g$, and such that $|A \cap [1,x]| \gg x^{1/h - ε}$. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a $g$ that satisfies $g_h(ε) \ll ε^{-1}$, improving the bound $g_h(ε) \ll ε^{-h+1}$ obtained by Vu. Finally we use the "alte…
Given a variety over $\mathbb{Q}$, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős-Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.
In 1972 Delange observed in analogy of the classical Erd\H os-Wintner theorem that $q$-additive functions $f(n)$ has a distribution function if and only if the two series $\sum f(d q^j)$, $\sum f(d q^j)^2$ converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the $q$-ary and Cantor case we focus on the Zeckendorf expansion that is based on the Fibonacci sequence, where we provide a sufficient and necessary condition for the existence of a distribution function, namely that the two …
In 1982, Erdös and van Lint showed an estimate for the average of the ratio of the smallest and largest prime divisor of $n$. In this note, we apply C.H. Jia's method to give an estimate for the average of positive integer power of the ratio.
Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subgroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erdös-Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that $\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of $\log G(n)$ and $\log I(n)$.
In this paper, we obtain sharp estimates for the number of lattice points under and near the dilation of a general parabola, the former generalizing an old result of Popov. We apply Vaaler's lemma and the Erdős-Turan inequality to reduce the two underlying counting problems to mean values of a certain quadratic exponential sums, whose treatment is subject to classical analytic techniques.
In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.
Let $β>1$, $I$ be the unite interval $[0,1)$ and $ϕ$ be an integer function defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leqϕ(n)\leq n$. Denote by $A_ϕ(x,β)$ the Erdös-Rényi average of $x\in I$ associated with the function $ϕ$ in $β$-expansion and $I_β$ the range of $A_ϕ(x,β)$ for $x\in I$. For the level set \begin{align*} ER_ϕ^β(α)=\left\{x\in I\colon A_ϕ(x,β)=α\right\},\quad\text{where}\ α\in I_β, \end{align*} in this paper we will determine its Hausdorff dimension under the assumption $ϕ(n)\to\infty$ as $n\to\infty$ and $ϕ$ is the integer part of some slowly varying sequence. Besides…
