Related papers: Distribution Laws of Smooth Divisors

The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for…

The index of codivisibility of a set of integers is the size of its largest subset with a common prime divisor. For large random samples of integers, the index of codivisibility is approximately normal.

We study concentration properties of random vectors of the form $AX$, where $X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. We show that the distribution of $AX$ is well spread in space whenever the…

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the…

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.

We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some…

We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable…

Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ``well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to…

We estimate the local laws of the distribution of the middle prime factor of an integer, defined according to multiplicity or not. An asymptotic estimate with effective remainder is provided for a wide range of values. In particular this…

Simple divisibility rules are given for the 1st 1000 prime numbers.

This paper deals with function field analogues of famous theorems of Laudau which counted the number of integers which have $t$ prime factors and R. Hall which researched the distribution of divisors of integers in residue classes.\;We…

A positive integer n is called r-free if n is not divisible by the r-th power of a prime. Generalizing earlier work of Orr, we provide an upper bound of Bombieri-Vinogradov type for the r-free numbers in arithmetic progressions.

An "element-free" probability distribution is what remains of a probability distribution after we forget the elements to which the probabilities were assigned. These objects naturally arise in Bayesian statistics, in situations where…

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 $\omega(s(n))$ has normal order $\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We…

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.

A phenomenological law, called Benford's law, states that the occurrence of the first digit, i.e., $1,2,...,9$, of numbers from many real world sources is not uniformly distributed, but instead favors smaller ones according to a logarithmic…

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.

In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.

We employ the spectral theory of Eisenstein series to prove that the Hardy sums, integer-valued analogs of the classical Dedekind sums, are uniformly distributed in $\mathbf{Z} / m \mathbf{Z}$ for any integer $m > 1$.

We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of…