International Journal of Number Theory最新文献

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The Barnes–Hurwitz zeta cocycle at s = 0 and Ehrhart quasi-polynomials of triangles s = 0 时的巴恩斯-赫尔维茨zeta 循环和三角形的埃尔哈特准多项式

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s179304212450057x

Milton Espinoza

Following a theorem of Hayes, we give a geometric interpretation of the special value at $s = 0$ of certain $1$ -cocycle on ${PGL}_{2} (ℚ)$ previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at $s = 0$ , a generalization and a new proof of Hayes’ theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in $ℝ^{2}$ .

根据海耶斯的一个定理，我们给出了作者之前介绍的 PGL2(ℚ)上某些 1 循环在 s=0 时的特殊值的几何解释。这项工作产生了三个主要结果：我们的 s=0 处的循环的明确公式，海耶斯定理的概括和新证明，以及ℝ2 中某些三角形的埃尔哈特准多项式的第零系数的优雅求和公式。

引用次数: 0

Linear algebra and congruences for MacMahon’s k-rowed plane partitions MacMahon k 行平面分区的线性代数和全等式

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s1793042124500702

Shi-Chao Chen

In this paper, we provide an algorithm to detect linear congruences of $p l_{k} (n)$ , the number of MacMahon’s $k$ -rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the $k$ -rowed plane partition functions. We also show $p (n, m)$ that the number of partitions at most $m$ parts always admits linear congruences.

在本文中，我们提供了一种检测 plk(n)（麦克马洪 k 行平面分区数）线性全等的算法，并给出了 k 行平面分区函数的拉马努金式全等不存在的定量结果。我们还证明了 p(n,m)，即最多有 m 个部分的分割数总是允许线性全等。

引用次数: 0

Fast computation of generalized dedekind sums 广义推演和的快速计算

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s179304212450060x

Preston Tranbarger, Jessica Wang

We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.

我们构建了一种算法，将计算广义戴德金和的复杂度从指数时间降低到多项式时间。为此，我们使用了群论中的高效词重写过程。

引用次数: 0

The Manin–Peyre conjecture for certain multiprojective hypersurfaces 某些多射超曲面的马宁-佩雷猜想

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s1793042124500623

Xiaodong Zhao

By the circle method, an asymptotic formula is established for the number of integer points on certain hypersurfaces within multiprojective space. Using Möbius inversion and the modified hyperbola method, we prove the Manin–Peyre conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for certain smooth hypersurfaces in the multiprojective space of sufficiently large dimension.

通过圆法，建立了多射空间内某些超曲面上整数点数的渐近公式。利用莫比乌斯反演法和修正双曲线法，我们证明了马宁-佩雷猜想，即在足够大维度的多射空间中，某些光滑超曲面上有界反锥高的有理点数的渐近行为。

引用次数: 0

Higher Mertens constants for almost primes II 几乎素数的更高默顿常量 II

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s179304212450088x

Jonathan Bayless, Paul Kinlaw, Jared Duker Lichtman

For $k \geq 1$ , let $ℛ_{k} (x)$ denote the reciprocal sum up to $x$ of numbers with $k$ prime factors, counted with multiplicity. In prior work, the authors obtained estimates for $ℛ_{k} (x)$ , extending Mertens’ second theorem, as well as a finer-scale estimate for $ℛ_{2} (x)$ up to ${(log x)}^{- N}$ error for any $N > 0$ . In this paper, we establish the limiting behavior of the higher Mertens constants from the $ℛ_{2} (x)$ estimate. We also extend these results to $ℛ_{3} (x)$ , and we comment on the general case $k \geq 4$ .

对于 k≥1，让ℛk(x)表示具有 k 个质因数的数到 x 的倒数和，以倍数计数。在之前的工作中，作者扩展了梅尔腾斯第二定理，得到了ℛk(x)的估计值，并对任意 N>0 的ℛ2(x)进行了更精细的估计，误差可达 (logx)-N。在本文中，我们从ℛ2(x) 估计中建立了较高默顿常量的极限行为。我们还将这些结果扩展到ℛ3(x)，并对 k≥4 的一般情况进行了评论。

引用次数: 0

Variations on a theorem of Capelli 卡佩利定理的变式

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s1793042124500465

Pradipto Banerjee

Elementary irreducibility criteria are established for <math altimg="eq-00002.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">)</mo></math> where <math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></math> is irreducible over <math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>ℚ</mi></math> and <math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>p</mi></math> is a prime. For instance, our main criterion implies that if <math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">)</mo></math> is reducible over <math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>ℚ</mi></math>, then <math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> divides <math altimg="eq-00009.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo stretchy="false">)</mo></math> modulo <math altimg="eq-00010.gif" display="inline" overflow="scroll"><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math>. Among several applications, it is shown that if <math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> has coefficients in <math altimg="eq-00012.gif" display="inline" overflow="scroll"><mo stretchy="false">{</mo><mo stretchy="false">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></math>, then <math altimg="eq-00013.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> is irreducible over <math altimg="eq-00014.gif" display="inline" overflow="scroll"><mi>ℚ</mi></math> excluding a couple of obvious exceptions. As another application, it is proved that if <math altimg="eq-00015.gif" display="inline" overflow="scroll"><mi>n</mi><mo>></mo><mn>4</mn></math> and <math altimg="eq-00016.gif" display="inline" overflow="scroll"><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo>

在 f(x)∈ℤ[x] 在ℚ上不可还原且 p 是素数的情况下，为 f(xp) 建立了基本的不可还原性准则。例如，我们的主要准则意味着，如果 f(xp) 在ℚ上是可还原的，那么 f(x) 除以 f(xp) modulo p2。在一些应用中，证明了如果 f(x) 的系数在{-1,1}中，那么 f(x2) 在ℚ上是不可还原的，但有几个明显的例外。另一个应用证明，如果 n>4 和 a1,a2,...,an 是不同的整数，那么对于𝜀∈{-1,1}，多项式 (x2-a1)(x2-a2)⋯(x2-an)+𝜀 在ℚ 上是不可约的，除非 n 是奇数且𝜀=-1。本文重点讨论了 f(0)∈{-1,1} 的非循环单项式 f(x)。在这些情况下，除其他外，还证明了如果 p≫(degf)logmax{2,H(f)}，其中 H(f) 表示 f(x) 的高，那么 f(xp) 在ℚ上是不可还原的。不可还原性标准的证明依赖于卡佩利关于 f(xm) 因式分解的一般结果。

引用次数: 0

A conjecture of Hegyvári 黑格瓦里的猜想

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-16 DOI: 10.1142/s1793042124500477

Xing-Wang Jiang, Wu-Xia Ma

For a given sequence $A$ of nonnegative integers, let $P (A)$ be the set of all finite subsequence sums of $A$ . $A$ is called complete if $P (A)$ contains all sufficiently large integers. A real number $α > 0$ is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of $α$ . Hegyvári conjectured that $A_{α, β}$ is complete if $α$ or $β$ is i.d.f. and $α / β \neq 2^{l} (l \in ℤ)$ , where $A_{α, β} = {[α], [β], \dots, [2^{n} α], [2^{n} β], \dots}$ is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.

对于给定的非负整数序列 A，让 P(A) 是 A 的所有有限子序列和的集合。如果 P(A) 包含所有足够大的整数，则称 A 为完全序列。如果数字 1 在 α 的二进制表示中出现无限多次，则实数 α>0 被称为无限二分数（简称 i.d.f.）。Hegyvári 猜想，如果 α 或 β 是 i.d.f.，且 α/β≠2l(l∈ℤ) ，则 Aα,β 是完全的，其中 Aα,β={[α],[β],...,[2nα],[2nβ],... } 是一个整数序列。本文给出了 Hegyvári 猜想的部分结果。

引用次数: 0

Values of certain Dirichlet series and higher derivative formulas of trigonometric functions 某些 Dirichlet 级数的值和三角函数的高导数公式

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-13 DOI: 10.1142/s1793042124500519

Dominic Lanphier, Allen Lin

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.

我们确定了某些 Dirichlet 级数和相关无穷级数的新值。这些公式扩展了多位学者的研究成果。为了得到这些结果，我们应用了三角函数高导数公式的最新展开式。我们还研究了这些级数值的超越性以及某些相关无穷级数值的算术关系。

引用次数: 0

On bounded basis with prescribed representation functions 在有界的基础上，用规定的表示函数

3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2023-11-02 DOI: 10.1142/s1793042124500179

Fang-Gang Xue

Let [Formula: see text] be the set of integers and [Formula: see text] the set of positive integers. For a nonempty set [Formula: see text] of integers and any integers [Formula: see text], [Formula: see text] with [Formula: see text], define [Formula: see text] as the number of solutions of [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] A set [Formula: see text] of integers is defined as a basis of order [Formula: see text] for [Formula: see text] if [Formula: see text] for every integer [Formula: see text]. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function [Formula: see text], there exists a bounded basis of order [Formula: see text] for [Formula: see text] such that [Formula: see text] for every integer [Formula: see text].

设[公式:见文]为整数集，[公式:见文]为正整数集。对于整数和任何整数的非空集合[公式:见文]，[公式:见文]与[公式:见文]，定义[公式:见文]作为[公式:见文]的解的个数，其中[公式:见文]和[公式:见文]对于[公式:见文]，整数集合[公式:见文]被定义为有序的基础[公式:见文]如果[公式:见文]对于每个整数[公式:见文]。2004年，Nešetřil和Serra考虑了一类有界基的Erdős-Turán猜想。本文推广了上述结果，得到:对于任意函数[公式:见文]，对于[公式:见文]存在一个阶[公式:见文]的有界基，使得[公式:见文]对于每一个整数[公式:见文]。

引用次数: 0

Infinite families of solutions for A3 + B3 = C3 + D3 and A4 + B4 + C4 + D4 + E4 = F4 in the spirit of Ramanujan 以拉马努扬精神为基础的 A3 + B3 = C3 + D3 和 A4 + B4 + C4 + D4 + E4 = F4 的无穷解族

3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2023-11-02 DOI: 10.1142/s1793042124500283

Archit Agarwal, Meghali Garg

Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler’s Diophantine equation [Formula: see text]. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation [Formula: see text] in the spirit of Ramanujan.

拉马努金在他丢失的笔记本中给出了一个有趣的恒等式，它可以生成欧拉丢芬图方程的无穷族解[公式:见原文]。在本文中，我们本着拉马努金的精神，对上述丢番图方程和丢番图方程[公式:见文]给出了几个无穷族的解。

引用次数: 0

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