William D. Banks, Igor E. Shparlinski
We study the function $Θ(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also indicate some cryptographic applications of our results.
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We establish an explicit inequality for the number of divisors of an integer $n$. It uses the size of $n$ and its number of distinct prime divisors.
We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs for some fixed values of $k$. In addition, we exhibit the run of 11 consecutive positive integers each with exactly 36 divisors, the run of 13 consecutive positive integers each with exactly 12 divisors, the run of 17 consecutive positive integers each with exactly 24 divisors, and the run of 17 consecutive positive integers each with exactly 48 divisors.
For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove a recursion for divisor function $σ_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ As a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function $σ_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$
The divisor function over integers with a missing digit
本文研究在具有数字限制的整数集合上除数函数的求和问题。
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.
Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q.
Let H(x,y,z) be the number of integers $\le x$ with a divisor in (y,z] and let H_1(x,y,z) be the number of integers $\le x$ with exactly one such divisor. When y and z are close, it is expected that H_1(x,y,z) H(x,y,z), that is, an integer with a divisor in (y,z] usually has just one. We determine necessary and sufficient conditions on y and z so that H_1(x,y,z) H(x,y,z). In doing so, we answer an open question from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223.
We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-γ})^{-1} = 2.280...$ and $γ$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.
A positive integer $n$ is called $φ$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $φ$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$.
Suppose that $θ$ is irrational. Then almost all elements $ν\in {\mathbb Z}[θ]$ that may be written as a $k$-fold product of the shifted integers $n+θ$ $(n\in {\mathbb N})$ are thus represented essentially uniquely.
Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1μ_1+a_2μ_2+\ldots+a_sμ_m$, where positive integers $μ_1,μ_2,\ldots,μ_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $μ_1,μ_2,\ldots,μ_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be writt…
For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.
We improve existing estimates of moments of the Riemann zeta function. As a consequence, we are able to derive new estimates for the asymptotic behaviour of $\sum_{N α\le x} \mathfrak{t}_k(α)$, where $N$ stands for the norm of a complex number and $\mathfrak{t}_k$ is the $k$-dimensional divisor function on Gaussian integers.
Let $\taue_k \colon \Z\to\Z$ be a multiplicative function such that $ \taue_k(p^a) = \sum_{d_1... d_k=a} 1 $. In the present paper we introduce generalizations of $\taue_k$ over the ring of Gaussian integers $\Zi$. We determine their maximal orders by proving a general result and establish asymptotic formulas for their average orders.
We have found maximum possible runs of consecutive positive integers each having exactly $k$ divisors for some fixed values of $k$. In addition, we exhibit the run of 10 consecutive positive integers each having exactly 12 divisors and two runs of 14 consecutive positive integers each having exactly 24 divisors.
The Vandermonde Determinant of the Divisors of an Integer
研究针对整数正因子有序序列定义的 Vandermonde 行列式型算术函数 $V(n)$，核心问题是对其进行估计。
Divisors of an Integer in a Short Interval
本研究定义并考察短区间内整数因数的计数函数，关注区间长度不超过起点时因数的分布。
On the greatest common divisor of integer parts of polynomials
本文受Bergelson和Richter的问题启发，研究了正整数与多项式整数部分构成互素元组的计数问题，并给出了误差项强度依赖于首项系数丢番图性质的渐近公式。
A classical result due to Deshouillers, Dress and Tenenbaum asserts that on average the distribution of the divisors of the integers follows the arcsine law. In this paper, we investigate the distribution of smooth divisors of the integers, that is, those divisors which are free of large prime factors. We show that on average these divisors are distributed according to a probability law that we will describe.
An Erdős-Kac theorem for integers with dense divisors
本文证明了连续因子之比有上界的大整数其素因子个数服从近似正态分布，并将该结果在两个不同方向上进行了推广。
Exponential sums over integers without large prime divisors
本文通过更精确地利用无大素因子整数的因数分解并结合Type I和Type II界，改进了此类整数及其 $ν$ 次幂上的指数和的现有界限。
On a Problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers
针对Erdős和Mirsky关于连续整数因数个数之比的问题，研究了满足存在无穷多整数$n$使$\alpha=\log\frac{d(n+1)}{d(n)}$的实数$\alpha$集合的闭包$\mathcal{L}$，并给出其密度的改进下界。
We determine, up to multiplicative constants, the number of integers $n\le x$ that have no prime factor $\le w$ and a divisor in $(y,2y]$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table which are free of prime factors $\le w$, and the number of distinct fractions of the form $\frac{a_1a_2}{b_1b_2}$ with $1\le a_1 \le b_1\le N$ and $1\le a_2\le b_2 \le N$.
The mean number of divisors for rough, dense and practical numbers
本文给出了无小素因子整数、连续约数比有界整数及实用数的约数平均个数的渐近估计，并证实了 Margenstern 的猜想。
We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.
A formula of counting divisors in integers rings: a generalization of the divisor function $d_0(n)$
本文通过建立整数环中理想结构与加性组合学理论之间的联系，证明了在具有循环类群的整数环中，每个非零理想均可分解为一个最大主因子和一个总长度受达文波特常数限制的理想乘积。通过这种分解，找到了理想I=(α, β)生成元的除数。本文的核心成果是利用有限阿贝尔群上的特征理论，推导出一个闭合公式，用于精确计数给定序列的零和子序列数目。该公式在建立主理想与零和序列之间的对应关系下，提供了对任意给定理想主理想除数的精确计数，从而计数理想I=(α, β)生成元的公共除数。这一结果构成了从唯一分解环到具有有限类群的任何迪克因环的除数函数d₀(n)的自然推广。最后，本文基于这些零和子序列的计数，对O_K中的不可约元进行了特征化。
We say a polynomial f having integer coefficients is strongly coefficient convex if the set of coefficients of f consists of consecutive integers only. We establish various results suggesting that the divisors of x^n-1 with integer coefficients have the tendency to be strongly coefficient convex and have small coefficients. The case where n=p^2*q with p and q primes is studied in detail.
Runs of Consecutive Integers Having the Same Number of Divisors
本文研究小于 N 且约数个数相同的连续整数最长段的长度上界，先用初等方法得到指数 λ=1/2，再利用 Jacobsthal 函数估计改进到 λ=1/3。
Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers
本文引入与正则除数系和任意正整数集相关的广义分圆多项式，证明其具有整数系数并可分解为经典分圆多项式的乘积，同时给出新的 Menon 型恒等式。
On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting
本文研究了与完美平方整数相关的除数函数的渐进行为，推导了特定求和式的渐近公式。
For each positive integer $N$, define $$S'_N \ =\ \{1 < d < \sqrt{N}: d|N\}\mbox{ and }L'_N \ =\ \{\sqrt{N} < d < N : d|N\}.$$ Recently, Chentouf characterized all positive integers $N$ such that the set of small divisors $\{d\le \sqrt{N}: d|N\}$ satisfies a linear recurrence of order at most two. We nontrivially extend the result by excluding the trivial divisor $1$ from consideration, which dramatically increases the analysis complexity. Our first result characterizes all positive integers $N$ such that $S'_N$ satisfies a linear recurrence of order at most two. Moreover, our second result ch…
Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.
By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.
A positive integer $n$ is called practical if all integers between $1$ and $n$ can be written as a sum of distinct divisors of $n$. We give an asymptotic estimate for the number of integers $\le x$ which have a practical divisor $\ge y$.
We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $ε>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - ε} \le z \le \min(y^{C},x^{1/2-ε}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key e…
Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $τ$ and $μ$ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{τ,k}(x)$ and $S_{μ,k}(x)$ for $k\geq 3$.
The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. In the paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors and exponentially coprime integers.
The $j$th divisor function $d_j$, which counts the ordered factorisations of a positive integer into $j$ positive integer factors, is a very well-known arithmetic function; in particular, $d_2(n)$ gives the number of divisors of $n$. However, the $j$th non-trivial divisor function $c_j$, which counts the ordered proper factorisations of a positive integer into $j$ factors, each of which is greater than or equal to 2, is rather less well-studied. We also consider associated divisor functions $c_j^{(r)}$, whose definition is motivated by the sum-over divisors recurrence for $d_j$. After reviewin…
A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}τ(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.
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.
Smoothed Shifted Convolutions of Generalised Divisor Functions
研究广义除数函数 \(d_k(n)\) 与除数函数 \(d(n)\) 的平滑移位卷积，证明其渐近公式，有与 \(k\) 无关的幂节省误差项，改进了特定条件下前人结果。
An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$, valid uniformly for $t\ge 2$.
It is well-known that for any distinct positive integers $k$ and $n$, the numbers $2^{2^k}+1$ and $2^{2^n}+1$ are relatively prime. In this paper we consider the situation when 1 is replaced by some positive integer $d>1$
The {\em Liouville function} is defined by $\gl(n):=(-1)^{Ω(n)}$ where $Ω(n)$ is the number of prime divisors of $n$ counting multiplicity. Let $\z_m:=e^{2πi/m}$ be a primitive $m$--th root of unity. As a generalization of Liouville's function, we study the functions $\gl_{m,k}(n):=\z_m^{kΩ(n)}$. Using properties of these functions, we give a weak equidistribution result for $Ω(n)$ among residue classes. More formally, we show that for any positive integer $m$, there exists an $A>0$ such that for all $j=0,1,...,m-1,$ we have $$#\{n\leq x:Ω(n)\equiv j (\bmod m)\}=\frac{x}{m}+O(\frac{x}{\log^A x…
Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors
本文证明了在素因子个数为d的正整数集合中，满足划分函数Newman猜想的正整数M的密度为1，并将该类猜想推广至弱全纯模形式，应用于t-core划分与广义Frobenius划分。
On a Smoothed Walfisz Divisor Problem
用除数函数σ替换Dirichlet除数函数τ，研究平滑Dirichlet除数问题，证明σ(n)加权和的显式渐近公式，从而消除经典平均阶中的困难部分，并推导Walfisz除数问题中误差项积分的收敛性。
Greatest common divisors of values of integer polynomials and an application to maximal tori
本文给出计算两个整系数多项式在所有整数点取值的最大公因数的方法，并将其用于确定例外李型群中全部极大环面的初等因子结构。
The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime divisors of some single polynomial. By combining this result with a theorem of Ax we get that for any system $F$ of multivariate polynomial equations with integer coefficients, the set of primes $p$ for which $F$ is solvable modulo $p$ is the set of prime divisors of some univariate polynomial. In addition, we prove results on the densities of the prime divisors of…
In a recent paper, we considered integers n for which the polynomial x^n - 1 has a divisor in Z[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial x^n-1 has a divisor in F_p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.
Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for $y+y/\log^C y < z\le y^2$ and $y\le x^{1-δ}$, showing that the order is the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$ with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and $δ>0$ is arbitrarily small.
Factorization of Dyck words and the distribution of the divisors of an integer
该研究探讨戴克词$\u27e8\negthinspace\u27e8 n \u27e9\negthinspace\u27e9_\u03bb$的不可约因式分解与整数$n$的除数分布的关联，并给出$\u03bb$-稠密可除数的刻画，拓展了数论与组合字理论的交叉联系。
We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q \le x^{2/3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.
Explicit and Effective Estimates for the error term in the Generalised Divisor Problem
本文对广义除数问题中误差项Δ_k(x)的误差项进行了显式和有效估计，特别是k在3到9之间的整数，改进了已知显式界限中的x幂次，并提供了推导所有k≥3整数误差项界限的方法。
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 extend the Selberg-Delange method to handle the following problems.\;The number of monic polynomials with degree $n$ have $t$ irreducible factors;\;The number of monic polynomials with degree $n$ in some residue classes have $t$ irreducible factors and the residue classes distribution of divisors of monic polynomial.\;
We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them belonging to a specified interval. In particular, this gives an estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.
Given a positive integer $n$ the $k$-fold divisor function $d_k(n)$ equals the number of ordered $k$-tuples of positive integers whose product equals $n$. In this article we study the variance of sums of $d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $d_k(n)$ in short intervals of certain lengths for $k=3$ and for $k \ge 4$ under the assumption of the Lindelöf hypothesis.
We study equations of the form $σ(p^{q-1})=Az$, where $p$ is a prime, $q$ is a fixed odd prime, $A$ is a fixed integer and $z$ is an integer composed of primes in a fixed finite set. We shall improve upper bounds for the size and the number of solutions of such equations.
Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.
Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
本文证明三素数整数平铺中存在无支撑分圆因子，刻画了平方周期情形，并给出最小模数及Apéry集构造。
Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural extension to negative integers $r.$ We give recurrence properties and explicit formulae for these novel arithmetic functions. Specifically, the functions $c_j^{(-j)}(n)$ count, up to a sign, the number of ordered factorisations of $n$ into $j$ square-…
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