Randell Heyman

activity

20132023

most citedHyperbolic summation for functions of the GCD and LCM of several integers

11 citations · 29 across the 16 of their papers we have counts for

collaborators

19 papers

math.NT2023

On some floor function sets

Randell Heyman, MD Rahil Miraj

Let $X$ be a positive integer and $t$ a real number great than 1. The family of sets ${⌊ \frac{X}{n ^{t}} ⌋ : 1 \leq n \leq X}$ have an interesting prim…

math.NT2023

Sparse sets that satisfy the prime number theorem

Olivier Bordellès, Randell Heyman, Dion Nikolic

We investigate various sparse sets that satisfy the prime number theorem. The sparsest of these sets, ${⌊ x / n^{t} ⌋ : n \leq x}$ , has density approaching $1/ x$ as $t$ appr…

math.NT2022★ 2 cited

Estimates for $k$ -dimensional spherical summations of arithmetic functions of the GCD and LCM

Randell Heyman, László Tóth

Let $k \geq 2$ be a fixed integer. We consider sums of type $\sum_{n_{1}^{2} + \dots + n_{k}^{2} \leq x} F (n_{1}, \dots, n_{k})$ , taken over the $k$ -dimensional spherical region $\{(n_1,\ldots,n_k)\i…

math.NT2022★ 11 cited

Hyperbolic summation for functions of the GCD and LCM of several integers

Randell Heyman, László Tóth

Let $k \geq 2$ be a fixed integer. We consider sums of type $\sum_{n_{1} \dots n_{k} \leq x} F (n_{1}, \dots, n_{k})$ , taken over the hyperbolic region $\{(n_1,\ldots,n_k)\in {\Bbb N}^k: n_1\cdo…

math.NT2021

Primes in floor function sets

Randell Heyman

Let $x$ be a positive integer. We give an asymptotic formula for the number of primes in the set $\{\fl{x/n}, 1 \le n \le x\}$ and give some related results.

math.NT2021

A summation of the number of distinct prime divisors of the lcm

Randell Heyman

Let $x$ be a positive integer. We give an asymptotic result for $ω (lcm (m, n))$ summed over all positive integers $m$ and $n$ with $mn \leq x$ . This answers an open que…

math.NT2020★ 9 cited

On certain sums of arithmetic functions involving the gcd and lcm of two positive integers

Randell Heyman, László Tóth

We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn \leq x} f ((m, n))$ and $\sum_{mn \leq x} f ([m, n])$ , where $f$ belongs to certain classes of ar…

math.NT2020

A summation involving the number of divisors function and the GCD function

Randell Heyman

Let $N$ be a positive number. We give an asymptotic formula for the sum of $τ (g cd (a, b))$ for all $a$ and $b$ with $ab \leq N$ .

math.NT2019

Cardinality of a floor function set

Randell Heyman

Fix a positive integer X. We quantify the cardinality of the set ${⌊ X / n ⌋}_{n = 1}^{X}$ . We discuss restricting the set to those elements that are prime, semiprime or si…

math.NT2018

On a sum involving the Euler function

Olivier Bordellès, Lixia Dai, Randell Heyman +2

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n ⩽ x} φ (⌊ x / n ⌋)$ , involving the Euler functions $φ$ and the integer…

math.NT2017

Tuples of polynomials over finite fields with pairwise coprimality conditions

Juan Arias de Reyna, Randell Heyman

Let $q$ be a prime power. We estimate the number of tuples of degree bounded monic polynomials $(Q_{1}, \dots, Q_{v}) \in (F_{q} [z])^{v}$ that satisfy given pairwise coprimality co…

math.NT2016

Evaluationally coprime linear polynomials

Randell Heyman

Two polynomials, $f, g \in Z [x]$ are evaluationally coprime at x if $g cd (f (x), g (x)) = 1$ . We give necessary and sufficient conditions for two such linear polynomials to have…

math.NT2015

Counting irreducible binomials over finite fields

Randell Heyman, Igor E. Shparlinski

We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.

math.NT2014★ 1 cited

Counting tuples restricted by coprimality conditions

Juan Arias de Reyna, Randell Heyman

Given a set $A = {(i_{1}, j_{1}), \dots, (i_{m}, j_{m})}$ we say that $(a_{1}, \dots, a_{v})$ exhibits pairwise coprimality if $g cd (a_{i}, a_{j}) = 1$ for all $(i, j) \in A$ . For a given positive $x$ we…

math.NT2014

On the Greatest Common Divisor of Shifted Sets

Randell Heyman, Igor E. Shparlinski

Given a set of $n$ positive integers ${a_{1}, \dots, a_{n}}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_{i}$ with $∣ h_{i} ∣ \leq H$ , $i =1,…

math.NT2013

On the number of polynomials of bounded height that satisfy Dumas's criterion

Randell Heyman

We study integer coefficient polynomials of fixed degree and maximum height $H$ , that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive up…

math.NT2013★ 4 cited

Pairwise non-coprimality of triples

Randell Heyman

We say that $(a_{1}, ..., a_{k})$ is pairwise non-coprime if $g cd (a_{i}, a_{j}) \neq = 1$ for all $1 \leq i < j \leq k$ . Let $a_{1}, a_{2}, a_{3}$ be positive integers less than $H$ . We obtain an asymptot…

math.NT2013

On Shifted Eisenstein Polynomials

Randell Heyman, Igor E. Shparlinski

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials.…

math.NT2013★ 2 cited

On the Number of Eisenstein Polynomials of Bounded Height

Randell Heyman, Igor E. Shparlinski

We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree $d$ and of height at most $H$ , as $H\to \inf…

On some floor function sets

Sparse sets that satisfy the prime number theorem

Estimates for k-dimensional spherical summations of arithmetic functions of the GCD and LCM

Hyperbolic summation for functions of the GCD and LCM of several integers

Primes in floor function sets

A summation of the number of distinct prime divisors of the lcm

On certain sums of arithmetic functions involving the gcd and lcm of two positive integers

A summation involving the number of divisors function and the GCD function

Cardinality of a floor function set

On a sum involving the Euler function

Tuples of polynomials over finite fields with pairwise coprimality conditions

Evaluationally coprime linear polynomials

Counting irreducible binomials over finite fields

Counting tuples restricted by coprimality conditions

On the Greatest Common Divisor of Shifted Sets

On the number of polynomials of bounded height that satisfy Dumas's criterion

Pairwise non-coprimality of triples

On Shifted Eisenstein Polynomials

On the Number of Eisenstein Polynomials of Bounded Height

Estimates for $k$ -dimensional spherical summations of arithmetic functions of the GCD and LCM