Melvyn B. Nathanson
Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets of positive integers. Let $g_{\mathcal{B} }(n)$ count the number of representations of $n$ in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots, h\}$. It is proved that $\liminf_{n\rightarrow \infty} g_{\mathcal{B} }(n) \geq 2$ implies $\limsup_{n\rightarrow \infty} g_{\mathcal{B} }(n) = \infty$.
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On the Erdős-Ginzburg-Ziv invariant and zero-sum Ramsey number for intersecting families
设$G$是有限阿贝尔群，且$m>0$具有$\exp(G)\mid m$ .设$s_{m}(G)$是广义Erdens-Ginzburg-Ziv不变量，它表示最小正整数$d$，使得$G$中长度为$d$的任何元素序列包含$G$中长度为$m$且和为零的子序列。对于任何整数$r>0$，令$\mathcal{I}_m^{(r)}$是所有$r$ -大小为$m$的均匀相交族的集合。设$R(\mathcal{I}_m^{(r)},G)$是最小正整数$d$，使得完全$r$ -均匀超图$K_{d}^{(r)}$的边的任何$G$ -染色都产生$\mathcal{I}_m^{(r)}$中某个相交族的零和拷贝.在其他结果中，我们主要证明了$Ω(s_{m}(G))-1\leq R (\mathcal{I}_{m}^{(r)}, \ G)\leq Ω(s_{m}(G)),$，其中$Ω(s_{m}(G))$表示最小正整数$n$，使${n-1 \choose r-1}\geq s_{m}(G)$，我们表明，如果$r\mid Ω(s_{m}(G))-1$然后$R (\mathcal{I}_{m}^{(r)}, \ G)= Ω(s_{m}(G))$。
Improved Ramsey bounds for generalized Schur equations
研究改进了广义施图姆方程的Ramsey界限，证明了在特定条件下，整数区间内的着色必然包含单色解，且给出了最优估计。
A structure theorem for multiplicative functions over the Gaussian integers and applications
证明了高斯整数上的乘性函数的一个结构定理，证明了高斯整数上的每一个有界乘性函数都可以分解为一个近似周期项和一个具有小U^{3}-Gowers一致范数的项.我们应用这一点来证明分区正则性的结果在高斯整数的某些方程涉及二次型在三个变量。例如，我们证明了对于高斯整数的任何有限着色，存在相同颜色的不同非零元素x和y，使得对于某个高斯整数n，x^{2}-y^{2}=n^{2}。这个语句在Z上的类比仍然是开放的。
A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erdős in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the lower order term: $π(n)+Θ(\frac{n^{3/4}}{(\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\{1,\ldots, n\}$ is $T(n)\cdot 2^{Θ(\frac{n^{3/4}}{(\log n)^{3/2}})}$ for a certain function $T(n)\approx 2^{1.815π(n)}$ which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determine…
In this note, we prove that under some conditions, certain products of integers related to Gauss factorials are always quadratic residues.
This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are $(2,8)$ and $(3,9)$. For triples, we prove the following theorem: If $a\notin \{2,8\}$ is a fixed integer larger than 1, then there are only finitely many triples $(a,b,c)$ of pairwise distinct integers larger than 1 such that $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ are each multiplicatively dependent. Moreover, these triple…
We establish an interesting upper bound for the moments of truncated Dirichlet convolution of Möbius function, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j \in \{1,\dots,n\}$. Also, we establish an estimate for the multiplicative energy of the set of divisors of an integer $n$.
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
两个研究得很好的拉姆齐理论问题考虑的子集的自然数，其中要么不包含三个元素的等差数列，或在几何级数。我们研究这个问题的推广，通过不同种类的进展，以避免和度量用于评估所得到的子集的密度。可以将3项等差数列看作序列$x, f_n(x), f_n(f_n(x))$，其中$f_n(x) = x + n$、$n$是非零整数。因此，避免三项等差数列等价于不包含具有$f_n \in\mathcal{F}_{\rm t}$的$x, f_n(x), f_n(f_n(x))$形式的三个元素，即整数平移的集合。人们可以类似地使用不同的函数族来构造相关的级数。我们调查了几个这样的家庭，包括几何级数（$f_n(x) = nx$与$n > 1$自然数）和指数级数（$f_n(x) = x^n$）。无级数集合通常被构造为“greenhouse”，包括每个数字，只要它不与前面的任何元素进行级数。Rankin用贪婪算术集刻画了贪婪几何级数自由集。我们刻画了贪婪指数集，证明了它的渐近密度为1，然后讨论了贪婪指数集的最优性如何依赖于用于定义级数的函数族。传统上，无进展集的大小是使用（上）渐近密度来测量的，但是我们考虑了几种不同的密度概念，包括均匀密度和指数密度。
On a problem of Arnold: the average multiplicative order of a given integer
对于g，n互质整数，设lg（n）表示g模n的乘法阶.受Arnold猜想的启发，我们研究了n <= x范围内与g互质且x趋于无穷大时l_g（n）的平均值。假设广义黎曼假设，我们表明，这个平均值基本上是一样大的平均卡迈克尔lambda函数。当p <= x时，我们还确定了l_g（p）的平均值在素数上的渐近性.
Multiplicative subsemigroups of the positive integers closed with respect to the number of digits
在本文中，我们提出了另一种方法来处理的问题Rosales等人。特别是，一旦一个基地$b$的整数的表示是固定的，我们描述了一个程序，用于构建最小的乘法子半群的正整数封闭的数字，其中包含一组正整数的数量。
We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent $1/k$: There are examples such that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+ε})$, for all $ε>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}öbius function'', J. Number Theory, 212 (2020), pp. 113--121.
In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.
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.
Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.
We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2πi/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. Also, if there exists an integer that cannot be written as the sum of an integer which is the norm of an ideal in in $\mathbb{Z}[\exp(2πi/p)]$ and at most $p-2$ powers of $p$, where $p\geq 3$ is a prime, then there are infinitely many such integers. Finally it is shown that there are infinitely many integers not the sum of the norm of an ideal…
Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of $2$ modulo $F_n$, for every positive integer $n$ not divisible by $3$, where $F_n$ denotes the $n$th Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$ and for all positive integers $n$ with $\gcd(a, F_n) = 1$. Our proof makes use of the so-called base-$φ$ expansion of real numbers.
In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).
On averages of completely multiplicative functions over co-prime integer pairs
本文研究了完全乘法函数在互质整数对上的平均值行为，推导了高斯整数上的Wirsing定理，并在多线性平均方面取得新进展。
In this paper, we first prove that given pairwise distinct algebraic numbers $α_1, \ldots, α_n$, the numbers $α_1+t, \ldots, α_n+t$ are multiplicatively independent for all sufficiently large integers $t$. Then, for a pair $(a,b)$ of distinct integers, we study how many pairs $(a+t,b+t)$ are multiplicatively dependent when $t$ runs through the integers. For such a pair $(a,b)$ with $b-a=30$ we show that there are $13$ integers $t$ for which the pair $(a+t,b+t)$ is multiplicatively dependent. We conjecture that $13$ is the largest value of such translations for any $(a,b)$, where $a \ne b$, pro…
For a wide class of sequences of integer domains $\mathcal{D}_n\subset\mathbb{N}^d$, $n\in\mathbb{N}$, we prove distributional limit theorems for $F(X_1^{(n)},\ldots,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and $(X_1^{(n)},\ldots,X_d^{(n)})$ is a random vector with uniform distribution on $\mathcal{D}_n$. As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set $\{X_1^{(n)},\ldots,X_d^{(n)}\}$. This generalizes previously known limit results for $\mathcal{D}_n$ being either a discrete cube or a discrete hyperbo…
Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the identity function.
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。
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