Research in Number Theory最新文献

英文中文

On the Iwasawa invariants of prime cyclotomic fields 素数分环场的Iwasawa不变量

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-04-11 DOI: 10.1007/s40993-023-00435-z

Sey Y. Kim

引用次数: 0

On the adjoint of higher order Serre derivatives 关于高阶Serre导数的伴随函数

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-04-11 DOI: 10.1007/s40993-023-00438-w

M. Charan

引用次数: 0

Identities associated to a generalized divisor function and modified Bessel function 广义除数函数和修正贝塞尔函数的恒等式

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-03-31 DOI: 10.1007/s40993-023-00431-3

D. Banerjee, B. Maji

引用次数: 4

Modular forms on $${{,textrm{SU},}}(2,1)$$ with weight $$frac{1}{3}$$ 模块形式在$${{,textrm{SU},}}(2,1)$$与重量 $$frac{1}{3}$$

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-03-28 DOI: 10.1007/s40993-022-00361-6

E. Freitag, Richard M. Hill

引用次数: 0

A note on Halász’s Theorem in Fq[t]documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {F}}_q[t]$$end{document} A note on Halász’s Theorem in Fq[t]documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {F}}_q[t]$$end{document}

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-03-22 DOI: 10.1007/s40993-023-00432-2

Ardavan Afshar

引用次数: 0

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I 有限域扩展中等均椭圆曲线非同构群结构的概率

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-01-22 DOI: 10.1007/s40993-023-00456-8

J. Cullinan, N. Kaplan

引用次数: 0

Unlikely intersections on the p-adic formal ball. p进正式球上不可能的交集。

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-01-01 DOI: 10.1007/s40993-023-00441-1

Vlad Serban

We investigate generalizations along the lines of the Mordell-Lang conjecture of the author's p-adic formal Manin-Mumford results for n-dimensional p-divisible formal groups $F$ . In particular, given a finitely generated subgroup $Γ$ of $F ({\bar{Q}}_{p})$ and a closed subscheme $X ↪ F$ , we show under suitable assumptions that for any points $P \in X (C_{p})$ satisfying $n P \in Γ$ for some $n \in N$ , the minimal such orders n are uniformly bounded whenever X does not contain a formal subgroup translate of positive dimension. In contrast, we then provide counter-examples to a full p-adic formal Mordell-Lang result. Finally, we outline some consequences for the study of the Zariski-density of sets of automorphic objects in p-adic deformations. Specifically, we do so in the context of the nearly ordinary p-adic families of cuspidal cohomological automorphic forms for the general linear group constructed by Hida.

我们研究了n维p可分形式群F的p进形式Manin-Mumford结果的莫德尔-朗猜想的推广。特别地，给定F (Q¯p)的有限生成子群Γ和闭子方案X“F”，在适当的假设下，我们证明了对于任意点p∈X (C p)满足n p∈Γ，对于某些n∈n，当X不包含正维的形式子群平移时，这些最小阶n是一致有界的。相反，我们提供了一个完整的p进形式莫德尔-朗结果的反例。最后，我们概述了研究p进变形中自同构对象集合的zariski密度的一些结果。具体来说，我们是在Hida构造的一般线性群的尖上同调自同构形式的近普通p进族的背景下进行的。

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引用次数: 2

Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang. 过配分函数的对数高阶差分不等式及王协章问题。

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-01-01 DOI: 10.1007/s40993-022-00420-y

Gargi Mukherjee

Let <math> <mrow><mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., <math> <mrow> <msup><mrow><mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mi>r</mi> <mo>-</mo> <mn>1</mn></mrow> </msup> <msup><mi>Δ</mi> <mi>r</mi></msup> <mo>log</mo> <mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> , by studying the inequality of the following form <dispformula> <math> <mrow> <mtable> <mtr> <mtd><mrow><mo>log</mo> <mrow><mo>(</mo></mrow> <mn>1</mn> <mo>+</mo> <mstyle> <mfrac><mrow><mi>C</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> <msup><mi>n</mi> <mrow><mi>r</mi> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn></mrow> </msup> </mfrac> </mstyle> <mo>-</mo> <mstyle> <mfrac><mrow><mn>1</mn> <mo>+</mo> <msub><mi>C</mi> <mn>1</mn></msub> <mrow><mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> </mrow> <msup><mi>n</mi> <mi>r</mi></msup> </mfrac> </mstyle> <mrow><mo>)</mo></mrow> <mrow></mrow></mrow> </mtd> <mtd><mrow><mrow></mrow> <mo><</mo> <msup><mrow><mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mi>r</mi> <mo>-</mo> <mn>1</mn></mrow> </msup> <msup><mi>Δ</mi> <mi>r</mi></msup> <mo>log</mo> <mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mrow></mrow> <mrow></mrow></mrow> </mtd> <mtd><mrow><mrow></mrow> <mo><</mo> <mo>log</mo> <mrow><mo>(</mo></mrow> <mn>1</mn> <mo>+</mo> <mstyle> <mfrac><mrow><mi>C</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> <msup><mi>n</mi> <mrow><mi>r</mi> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn></mrow> </msup> </mfrac> </mstyle> <mrow><mo>)</mo></mrow> <mspace></mspace> <mtext>for</mtext> <mspace></mspace> <mi>n</mi> <mo>≥</mo> <mi>N</mi> <mrow><mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> <mo>,</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <math><mrow><mi>C</mi> <mrow><mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> <mo>,</mo> <msub><mi>C</mi> <mn>1</mn></msub> <mrow><mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> <mo>,</mo> <mtext>and</mtext> <mspace></mspace> <mi>N</mi> <mrow><mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> </mrow> </math> are computable constants depending on the positive integer r, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of <math> <mrow> <msup><mrow><mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mi>r</mi> <mo>-</mo> <mn>1</mn></mrow> </msup> <msup><mi>Δ</mi> <mi>r</mi></msup> <mo>log</mo> <mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> than 0. By settling the problem, we are able to show that <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><munder><mo>lim</mo> <mrow><mi>n</mi> <mo>→</mo> <

让¯p (n) denote《overpartition功能。在这篇文章里,我们主要的目标是需要研究的有限的分歧asymptotic社会行为》《overpartition logarithm功能,神盾局(- 1)r - 1Δr p¯日志不平等》(n),由studying跟踪日志表格(1 + C (r) n r - 1 / 2 - 1 + C (r) n r ) ( - 1) r - 1Δr p¯日志(n ) log (1 + C (r) n r - 1 / 2)为n≥n (r ) , 在C (r)、C (r)N (r)经常依赖于积极的英特尔r，有决心的excitly。这个solves a posed问题由王,谢》和《张》背景下束缚在寻找一个更好的(- 1)r - 1Δr p¯日志(n)比0 - 9。settling偏难题,我们能干展示的lim) n→∞(- 1)r - 1Δr p¯日志(n) =π2 (1)r - n 1 2 r。。

引用次数: 0

Summing μ ( n ) : a faster elementary algorithm. 求和μ (n):一个更快的初等算法。

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-01-01 DOI: 10.1007/s40993-022-00408-8

Harald Andrés Helfgott, Lola Thompson

We present a new elementary algorithm that takes $time O_{ϵ} (x^{\frac{3}{5}}, {(log x)}^{\frac{8}{5} + ϵ}) and space O (x^{\frac{3}{10}}, {(log x)}^{\frac{13}{10}})$ (measured bitwise) for computing $M (x) = \sum_{n \leq x} μ (n),$ where $μ (n)$ is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $O (x^{1 / 5} {(log x)}^{5 / 3})$ by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of $x^{3 / 5} {(log x)}^{2} log log x$ .

我们提出了一种新的初等算法，用于计算M (x) =∑n≤x μ (n)，该算法花费时间O λ x 35 (log x) 85 + λ和空间O λ 3 10 (log x) 13 10(按位测量)，其中μ (n)是Möbius函数。这是1985年以来初等算法对x指数的第一次改进。我们还表明，通过使用(Helfgott in: Math computer 89:333- 350,2020)，可以将空间消耗减少到O (x 1 / 5 (log x) 5 / 3)，代价是让时间上升到x 3 / 5 (log x) 2 log log x的数量级。

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引用次数: 0

Geometric sieve over number fields for higher moments. 在数字域上进行几何筛选以获得更高的矩。

IF 0.8 Q3 MATHEMATICS

Research in Number Theory

Pub Date : 2023-01-01 Epub Date: 2023-08-02 DOI: 10.1007/s40993-023-00466-6

Giacomo Micheli, Severin Schraven, Simran Tinani, Violetta Weger

The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field K. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.

密度的几何筛是Poonen和Stoll（以及Ekedahl独立提出的）提出的一种非常方便的工具，用于计算给定整数子集的密度。在本文中，我们提供了一个有效的准则来寻找数域的代数整数环上有限维自由模的子集的密度的所有高阶矩（例如均值、方差）。更准确地说，我们提供了一个几何筛，它允许在一般数域K上计算与密度相对应的所有更高矩。这项工作以两种方式推进了对密度计算的几何筛的理解：一方面，它扩展了Bright、Browning和Loughran的结果，在那里他们提供了数域上密度的几何筛；另一方面，它将最近关于整数上期望值的几何筛的结果推广到代数整数环和高于期望值的矩。为了证明我们的方法的有效性和适用性，我们计算了数域上艾森斯坦多项式和移位艾森斯坦多项式的密度、均值和方差。这扩展了（并完全覆盖了）通过特殊方法获得的文献中的结果。

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引用次数: 0

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