Related papers: Distribution Laws of Smooth Divisors
We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on…
A hypothesis is put forward regarding the function $\pi_2(x)$ which describes the distribution of twin primes in the set of natural numbers. The function $\pi_2(x)$ is tested by evaluation and an empirical $\pi_2^{\ast}(x)$ is arrived at,…
In this note we consider the distribution of values of weighted sums of the von Mangoldt arithmetical function. By using a formula for the distribution of values of trigonometric polynomials, we are able to present evidence supporting the…
In recent work by Botkin, Dawsey, Hemmer, Just and the present author, a deterministic model of prime number distribution is developed based on properties of integer partitions that gives almost exact estimates for $\pi(n)$, the number of…
The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be $[y',y]$-smooth if all of its prime factors belong to the interval $[y',y]$. We identify suitable weights…
Consider $n$ independent random numbers with a uniform distribution on $[0,1]$. The number of them that exceed their mean is shown to have an Eulerian distribution, i.e., it is described by the Eulerian numbers. This is related to, but…
Benford's Law describes the prevalence of small numbers as the leading digits of numbers in many sets of integers. We prove a variant of Benford's law for many positive-density subsets of the primes. This follows from a more general result…
A discrete-time stochastic process derived from a model of basketball is used to generalize any discrete distribution. The generalized distributions can have one or two more parameters than the parent distribution. Those derived from…
We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.
In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and…
We apply recent bounds of the author (math.PR/0609224) for generalized Smirnov statistics to the distribution of integers whose prime factors satisfy certain systems of inequalities.
We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\bold M$…
Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal…
This article presents an efficient algorithm to generate a discrete uniform distribution on a set of $p$ elements using a biased random source for $p$ prime. The algorithm generalizes Von Neumann's method and improves computational…
Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…
In this paper we study the divisibility and primality properties of the Bernoulli random walk. We improve or extend some of our divisibility results to wide classes of iid or independent non iid random walks. We also obtain new primality…
We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of…
We study the average value of the divisor function $\tau(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$.…
Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le…
Given a probability distribution P, what is the minimum amount of bits needed to store a value x sampled according to P, such that x can later be recovered (except with some small probability)? Or, what is the maximum amount of uniform…