Randell Heyman
Let $x$ be a positive integer. We give an asymptotic result for $ω(\operatorname{lcm}(m,n))$ summed over all positive integers $m$ and $n$ with $mn \le x$. This answers an open question posed in a recent paper.
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Hyperbolic summation involving the function $Ω(n)$ and lcm
本文研究了在双曲区域 $\{(a,b,c) \in \mathbb{Z}_{\geq 1}^3 : abc \leq x\}$ 上涉及素因子计数函数 $Ω(n)$ 与最小公倍数的求和 $\sum_{abc \leq x} Ω([a,b,c])$，并为其推导出了渐近公式。
For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger than 1. Utilizing a representation of the natural numbers as an order-theoretical ring of prime power sets, this conjecture is shown to be equivalent to Frankl's union-closed sets conjecture. Some results for cases where the conjecture, which also has meaningful interpretations in graph and lattice theory, is known to hold are provided. An equivalent dual vers…
Hyperbolic summation for functions of the GCD and LCM of several integers
本文研究了多个整数最大公约数与最小公倍数函数在双曲区域上的求和问题，推导了带有余项的渐近公式，并推广了作者先前在二元情形下的结果。
By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.
A Voronoi summation formula for the shifted triple divisor function
本文证明了由加性特征扭曲的移位3重除数函数的Voronoi求和公式，主要工具是提供了移位$GL(3)$ Estermann函数的函数方程。
Let $N$ be a positive number. We give an asymptotic formula for the sum of $τ(\gcd(a,b))$ for all $a$ and $b$ with $ab \le N$.
We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$ belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$ denoting the gcd and lcm of the integers $m,n$. In particular, we investigate the functions $f(n)=τ(n), \log n, ω(n)$ and $Ω(n)$. We also define a common generalization of the latter three functions, and prove a corresponding result.
Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.
This paper is devoted to the study of statistical properties of the greatest common divisor and the least common multiple of random samples of positive integers.
We study the distribution of the generalized gcd and lcm functions on average. The generalized gcd function, denoted by $(m,n)_b$, is the largest $b$-th power divisor common to $m$ and $n$. Likewise, the generalized lcm function, denoted by $[m,n]_b$, is the smallest $b$-th power multiple common to $m$ and $n$. We derive asymptotic formulas for the average order of the arithmetic, geometric, and harmonic means of $(m,n)_b$. Additionally, we also deduce asymptotic formulas with error terms for the means of $(n_1,n_2,\cdots, n_k)_b$, and $[n_1,n_2,\cdots, n_k]_b$ over a set of lattice points, th…
Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM
研究了$k$维球面区域内涉及整数GCD和LCM算术函数的求和问题，推导了带余项的渐近公式。
In this note, we establish a Voronoï--Oppenheim summation formula for divisor functions over an arbitrary number field.
Character analogues of Cohen type identities and related Voronoi summation formulas
本文针对 Cohen 型恒等式给出特征类比，推导相关 Voronoi 求和公式，并证明实原特征下 L(1,χ) 的正性。
Given two sets $A$ and $B$ of integers, we consider the problem of finding a set $S \subseteq A$ of the smallest possible cardinality such the greatest common divisor of the elements of $S \cup B$ equals that of those of $A \cup B$. The particular cases of $B = \emptyset$ and $\#B = 1$ are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Minimum Cover Problem.
We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.
Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n<x\} \sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely, $\operatorname{lcm} \{f(p)\mid p<x\}$ where $p$ denotes a prime and obtain non-trivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of $f(p).$
The function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with large discriminant. This method is based on using LLL-reduced bases, the "jump algorithm" and Poisson summation formula.
We obtain asymptotics with a power saving error term for $\int_0^1|\sum_{n\le X} d(n)e(nα)|^sdα$ for $s < 2$, where $d(n) = \sum_{d | n} 1$ is the divisor function. We also obtain such asymptotics for $\int_0^1|\sum_{n\le X}λ_f(n)e(nα)|^sdα$ for all $s > 0$. Our proof an iterative method with repeated applications of Jutila's variant of the circle method and Voronoi summation.
We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form $u_n = a n (n + t) + b$ with $(a, t, b) \in {\mathbb{Z}}^3, a \geq 5, t \geq 0, \rm{gcd}(a, b) = 1$. From this, we deduce for instance the lower bound: $\mathrm{lcm}\{1^2 + 1, 2^2 + 1, ..., n^2 + 1\} \geq 0,32 (1,442)^n$ (for all $n \geq 1$). In the last part…
For integers $x$ and $y$, $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$ respectively. Denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$) as its $(i,j)$-entry. For any $x\in S$, define $G_{S}(x):=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S) \Rightarrow y\in \{d,x\}\}$. In this paper, we show th…
Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\widehat{\mathbb{A}}$ be the set of integers of $\mathbb{K}$ which are primitive over $\mathbb{Q}$ and $I(\mathbb{K})$ be its index. Gunji and McQuillan defined the following integer $i(\mathbb{K})=\underset{θ\in \widehat{\mathbb{A}}}{\text{lcm}}\;i(θ)$, where $i(θ)=\underset{x\in \mathbb{Z}}{\text{gcd}}\;F_θ(x)$ and $F_θ(x)$ is the characteristic polynomial of $θ$ over $\mathbb{Q}$. We prove that if $p$ is a prime number less than or equal to $n$ then there exists a number field $\mathbb{K}$ of degree $n$ for which $p$ d…
Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$
本文研究广义除数函数 $σ_{z}^{(k)}(n)$，在其 Dirichlet 级数不落入经典多 gamma 因子 Hecke 函数方程理论的情况下，建立显式 Voronoï 求和公式，并分析新型 Bessel 型核及其关键关系。
Gcd-closed sets and divisibility among power GCD matrices and power LCM matrices
本文证明了Hong猜想，即在满足特定条件的gcd闭集上，幂GCD矩阵和幂LCM矩阵在a|b时能被幂GCD矩阵整除。
In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.
In this article, we study a divisor function in an arbitrary number field akin to Koshliakov's work on Vorono\"{\dotlessi} summation formula. More precisely, we generalize Koshliakov's kernel and Koshliakov's transform over any number field to obtain identities for the Lambert series associated to the divisor function in an arbitrary number field.
Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on the divisible factor of this integer, that is: For any positive integer $n~(n>2)$, if there exists an integer $m$, such that $m|n~( 1 < m < n )$, then $n=\sum_{i=1}^m p_i $, where $ p_i~(i=1,2 ,3...m)$ is prime number. In addition, for more prime summands, the combinatorial counting is also discussed. For some special cases, some brief proofs are given. By the…
Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$
本文研究了所有正整数四元组(m,a,b,c)的问题，其中a≥b≥c，使得存在一个具有最小模m和模数的最小公倍数为2^a 3^b 5^c的覆盖系统。当m=2,3,4,5或6时，除了m=6且b=c=1的情况外，得到了所有这样的四元组的完整描述。还证明了如果模数的最小公倍数仅包含2，3或5作为质因数，则m≤9，并构造了一个具有m=8，a=8，b=3，c=2的覆盖系统。当存在覆盖系统时，提供了示例。通过整数规划或新的估计方法证明了覆盖系统的不存在性。
A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities
本文研究基-M到基-N移位映射所得序列b(n)的算术性质，证明其最终周期性条件并界定最小周期，在条件不满足时为Prouhet-Tarry-Escott问题提供新解，并引入有限差分恒等式扩展经典数字和恒等式。
The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $σ_{\mathbb{K},α}(n)$. The Dirichlet series associated to this function is $ζ_{\mathbb{K}}(s)ζ_{\mathbb{K}}(s-α)$. We give an expression for the Riesz sum associated to $σ_{\mathbb{K},α}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $α=0$, gives the Vorono\"ı summation formula associated to the d…
Kernel-Checked Exclusions for the Erdős-Selfridge Odd Covering Problem: Any Odd Covering of $\mathbb{Z}$ Has lcm Exceeding 10000
研究埃尔德什 - 塞尔弗里奇奇数覆盖问题，通过Lean 4形式化证明，利用密度论证、丰富度下限、中国剩余容量证书及枚举等方法，得出整数集\(\mathbb{Z}\)的奇数覆盖其模最小公倍数超10000的结论，贡献在于提供内核检查的认知成果。
Let $n\ge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_{n}(x)=\sum_{i=0}^n\frac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $\Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $\E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$ with all $c_i \ (0\le i\le n-1)$ being integers is irreducible over $\Q$ if either …
Divisibility among power GCD and power LCM matrices on certain gcd-closed sets
本文研究gcd闭集上幂GCD与幂LCM矩阵的整除性，证明满足特定条件时三类整除关系成立，还找到不满足该条件仍成立的集合，推广了已有定理并部分证实相关猜想。
Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities
本研究推导了刘维尔函数、莫比乌斯函数及除数函数平方的Voronoi求和公式，得到多类恒等式特例，并在相关数论猜想假设下证明了对应Riesz和及其误差项的振荡性质。
We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -- each is vitiated by divergent series. We concentrate on only one of the two incorrect "identities", which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of …
The Number of Subgroups and Cyclic Subgroups of Finite Group and Its Application by GAP Program
该研究提出计算有限群子群（重点为循环子群）的新方法，推导了双循环群与循环群直积的子群数公式，并通过GAP程序找出满足特定循环子群数条件的该类直积群。
The Normal Form of Smith's Matrices
本文研究Smith矩阵（GCD矩阵）的对角化，提出定理4.1和4.2，将整除性结果推广到满足条件$\mathcal{G}$的gcd封闭集。
Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind
本文研究由第一类无符号斯特林数生成的数值半群，通过gcd滤链和因子支撑性质证明其约化生成序列为望远镜结构，从而得到Apéry集矩形、显式Frobenius公式及自由对称性。
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