Related papers: On the least common multiple of several random int…

We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for…

Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed…

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that…

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of…

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with…

This thesis is devoted to studying estimates of the least common multiple of some integer sequences. Our study focuses on effective bounding of the $\mathrm{lcm}$ of some class of quadratic sequences, as well as arithmetic progressions and…

In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a…

In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly…

We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of NC(n). As a consequence, we show that the even generating function for irreducible meandric systems is…

Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with…

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and…

For each $n$, let $X_n \in \{0,\ldots,n\}$ be a random variable with mean $\mu_n$, standard deviation $\sigma_n$, and let \[ P_n(z) = \sum_{k=0}^n \mathbb{P}( X_n = k) z^k ,\] be its probability generating function. We show that if none of…

Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let…

This paper explores the use of "generated sets" $\{ \{ k \boldsymbol{\zeta} \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent…

Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|\xi_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n} z^k, $$ where…

We elaborate the notions of Martin-L\"of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform…

We study the singular values (and Lyapunov exponents) for products of $N$ independent $n\times n$ random matrices with i.i.d. entries. Such matrix products have been extensively analyzed using free probability, which applies when $n\to…

Landau function $g(n)$ is the maximal possible least common multiple of several positive integers with sum not exceeding $n$. Under additional assumptions that these numbers are the differences of disjoint bi-infinite arithmetic…

Given an integer $n \ge 2$, its prime factorization is expressed as $n= \prod_{i=1}^s p_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to…

Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{\omega(n)}$ is its unitary analogue, where $\omega(n)$ is the number of…