International Journal of Number Theory最新文献

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Variance of primes in short residue classes for function fields 函数域短残差类中的素数方差

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-26 DOI: 10.1142/s1793042124500763

Stephan Baier, Arkaprava Bhandari

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus $Q$ is a polynomial in $𝔽_{q} [T]$ such that $Q (0) \neq 0$ . The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int.Math.Res. Not.2014(1) (2014) 259-288]导出了函数场设置中算术级数和短区间中素数方差的渐近公式。在此，我们考虑计算算术级数和短区间交集中素数方差的混合问题。Keating 和 Rudnick 使用内卷将短区间转化为算术级数。我们沿用了他们的方法，但在算术级数中也应用了这种反卷。当模数 Q 是𝔽q[T]中的多项式时，Q(0)≠0，这样就产生了对偶算术级数。后者是我们在本文中始终保留的限制条件。最后，我们将讨论如何放宽这一条件。

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引用次数: 0

Multiplier systems for Siegel modular groups 西格尔模块群的乘法系统

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-26 DOI: 10.1142/s1793042124500684

Eberhard Freitag, Adrian Hauffe-Waschbüsch

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus $g > 1$ must be integral or half integral. Actually he proved that for a system $v (M)$ of complex numbers of absolute value 1

$\begin{array}{l} v (M) det {(C Z + D)}^{r} (r \in ℝ) (0.1) \end{array}$

can be an automorphy factor only if $2 r$ is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann.159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for $S L_{n} (n \geq 3)$ and $S p_{2 n} (n \geq 2)$ , Publ. Math. Inst. Hautes Études Sci.33 (1967) 59–137] of Bass–Milnor–Serre.

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad.Sci. Paris287 (1978) 203-208] 中证明的。(另见 7.1 [R.Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds.H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp.实际上，他证明了对于绝对值复数系统 v(M) 1v(M)det(CZ+D)r(r∈ℝ)(0.1) 只有当 2r 是积分时才能成为自形因子。我们给出了一个不同的证明。它使用了门尼克的结果[Zur Theorie der Siegelschen Modulgruppe, Math. Ann.159 (1965) 115-129]，即西格尔模群的有限指数子群是全等子群，以及[Solution of the congruence subgroup problem for SLn(n≥3) and Sp2n(n≥2), Publ.Math.高等科学研究所，33 (1967) 59-137]的 Bass-Milnor-Serre.

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引用次数: 0

The minimal odd excludant and Euler’s partition theorem 最小奇数不等式和欧拉分割定理

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-26 DOI: 10.1142/s1793042124500714

Andrew Y. Z. Wang, Zheng Xu

In this work, we establish two interesting partition identities involving the minimal odd excludant, which has attracted great attention in recent years. In particular, we find a strong refinement of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts.

在这项工作中，我们建立了两个涉及最小奇数不等式的有趣的分割等式，该等式近年来引起了极大的关注。特别是，我们发现了欧拉著名定理的有力改进，即把一个整数分割成奇数部分的个数等于把该整数分割成不同部分的个数。

引用次数: 0

Reciprocity formulae for generalized Dedekind–Rademacher sums attached to three Dirichlet characters and related polynomial reciprocity formulae 附加于三个 Dirichlet 字符的广义 Dedekind-Rademacher 和的互易公式及相关多项式互易公式

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-26 DOI: 10.1142/s1793042124500726

Brad Isaacson

We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck & Kohl, and the present author.

我们定义了霍尔、威尔逊和扎吉尔引入的广义戴德金-拉德马赫和的三字符类似物，并证明了它的互易公式，其中包含了文献中所有作为特例的附于（和不附于）德里赫特字符的广义戴德金-拉德马赫和的互易公式。此外，我们还证明了相关的多项式互易公式，这些公式包含了文献中作为特例的所有多项式互易公式，如 Carlitz、Beck & Kohl 和本文作者给出的公式。

引用次数: 0

On almost-prime k-tuples 关于几乎是素数的 k 元组

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-26 DOI: 10.1142/s1793042124500751

Bin Chen

Let $τ$ denote the divisor function and $ℋ = {h_{1}, \dots, h_{k}}$ be an admissible set. We prove that there are infinitely many $n$ for which the product $\prod_{i = 1}^{k} (n + h_{i})$ is square-free and $\sum_{i = 1}^{k} τ (n + h_{i}) \leq ⌊ ρ_{k} ⌋$ , where $ρ_{k}$ is asymptotic to $\frac{2126}{2853} k^{2}$ . It improves a previous result of Ram Murty and Vatwani, replacing $3 / 4$ by $2126 / 2853$ . The main ingredients in our proof are the higher rank Selberg sieve and Irving–Wu–Xi estimate for the divisor function in arithmetic progressions to smooth moduli.

让 τ 表示除数函数，ℋ={h1,...,hk} 是可容许集合。我们证明，有无穷多个 n 的积∏i=1k(n+hi)是无平方差的，且∑i=1kτ(n+hi)≤⌊ρk⌋，其中ρk 渐近于 21262853k2。它改进了拉姆-穆蒂和瓦特瓦尼之前的一个结果，用 2126/2853 取代了 3/4。我们证明的主要内容是光滑模的算术级数中的高阶塞尔伯格筛和除数函数的欧文-吴-西估计。

引用次数: 0

On the Diophantine equation σ2(X¯n) = σn(X¯n) 关于 Diophantine 方程 σ2(X¯n) = σn(X¯n)

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-26 DOI: 10.1142/s1793042124500635

Piotr Miska, Maciej Ulas

In this paper, we investigate the set $S (n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of $σ_{2} ({\bar{X}}_{n})$ and $σ_{n} ({\bar{X}}_{n})$ together with the biggest value of the variable $x_{n}$ appearing in the solution. Moreover, we enumerate all solutions for $n \leq 16$ and discuss the set of values of $x_{n} / x_{n - 1}$ over elements of $S (n)$ .

在本文中，我们研究了标题 Diophantine 方程的正整数解集 S(n)。特别是，对于给定的 n，我们证明了解的有界性，给出了 σ2(X¯n)和 σn(X¯n)的公共值以及解中出现的变量 xn 的最大值的精确上限。此外，我们列举了 n≤16 的所有解，并讨论了 S(n) 元素上 xn/xn-1 的值集。

引用次数: 0

Fourier coefficients of cusp forms on special sequences 特殊序列上尖顶形式的傅里叶系数

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-21 DOI: 10.1142/s1793042124500568

Weili Yao

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms $f$ and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.

在本文中，我们研究了在具有固定数目的不同素除数的整数上，原始尖顶形式 f 的归一化傅里叶系数的平方及其对称提升，并给出了它们在短区间内的渐近公式。

引用次数: 0

Density questions in rings of the form 𝒪K[γ] ∩ K 形式为 𝒪K[γ] ∩K 的环中的密度问题

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-21 DOI: 10.1142/s1793042124500581

Deepesh Singhal, Yuxin Lin

We fix a number field <math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>K</mi></math> and study statistical properties of the ring <math altimg="eq-00004.gif" display="inline" overflow="scroll"><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo stretchy="false">∩</mo><mi>K</mi></math> as <math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>γ</mi></math> varies over algebraic numbers of a fixed degree <math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>n</mi><mo>≥</mo><mn>2</mn></math>. Given <math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>k</mi><mo>≥</mo><mn>1</mn></math>, we explicitly compute the density of <math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>γ</mi></math> for which <math altimg="eq-00009.gif" display="inline" overflow="scroll"><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo stretchy="false">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mi>k</mi><mo stretchy="false">]</mo></math> and show that this does not depend on the number field <math altimg="eq-00010.gif" display="inline" overflow="scroll"><mi>K</mi></math>. In particular, we show that the density of <math altimg="eq-00011.gif" display="inline" overflow="scroll"><mi>γ</mi></math> for which <math altimg="eq-00012.gif" display="inline" overflow="scroll"><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">[</mo><mi>γ</mi><mo stretchy="false">]</mo><mo stretchy="false">∩</mo><mi>K</mi><mo>=</mo><msub><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msub></math> is <math altimg="eq-00013.gif" display="inline" overflow="scroll"><mfrac><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfrac></math>. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define <math altimg="eq-00014.gif" display="inline" overflow="scroll"><mi>X</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></math> to be a certain finite subset of <math altimg="eq-00015.gif" display="inline" overflow="scroll"><mstyle><mtext>Spec</mtext></mst

我们固定一个数域 K，研究当 γ 在固定度 n≥2 的代数数上变化时，环 𝒪K[γ]∩K 的统计性质。给定 k≥1，我们明确计算了 γ 的密度，其中 𝒪K[γ]∩K=𝒪K[1/k]，并证明它不依赖于数域 K。特别是，我们证明了 γ 的密度，其中 𝒪K[γ]∩K=𝒪K 是 ζ(n+1)ζ(n)。在最近的一篇论文 [Singhal and Lin, Primes in denominators of algebraic numbers, Int.J. Number Theory (2023), doi:10.1142/S1793042124500167] 中，作者定义 X(K,γ) 为 Spec(𝒪K) 的某个有限子集，并证明 X(K,γ) 决定了环𝒪K[γ]∩K。我们证明，如果𝔭1,𝔭2∈Spec(𝒪K)满足𝔭1∩≠𝔭2∩ℤ，那么事件𝔭1∈X(K,γ)和𝔭2∈X(K,γ)是独立的。当 t→∞ 时，我们研究|X(K,γ)|=t 时 γ 密度的渐近线。

引用次数: 0

Multi-partition analogue of q-binomial coefficients q 次二项式系数的多分区类似物

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s1793042124500659

Byungchan Kim, Hayan Nam, Myungjun Yu

We introduce the multi-Gaussian polynomial $G_{k} (M, N)$ , a multi-partition analogue of the Gaussian polynomial (also known as $q$ -binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of $G_{k} (M, N)$ . We also derive a Sylvester-type identity and its application.

我们介绍了多高斯多项式 Gk(M,N)，它是高斯多项式（又称 q-二项式系数）的多分区类似物，是某些受限多色分区的生成函数。我们研究了多高斯多项式的基本性质和 Gk(M,N) 的非对称性质。我们还推导出了一个西尔维斯特式特性及其应用。

引用次数: 0

Corrigendum to “The discriminant of compositum of algebraic number fields” 对 "代数数域集合的判别式 "的更正

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-03-20 DOI: 10.1142/s1793042124500489

Sudesh Kaur Khanduja

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.

我们指出[The discriminant of compositum of algebraic number fields, Int. J Number Theory15 (2019)] 中定理 1.1 的证明有误。J. Number Theory15 (2019) 353-360]中的定理 1.1 的证明有误。我们还证明了该定理的结果在附加假设的情况下成立。然而，该定理的结果在一般情况下是否成立还是一个悬而未决的问题。

引用次数: 0

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