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On a conjecture of Graham on the -divisibility of central binomial coefficients 格雷厄姆关于中心二项式系数可分性的猜想
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-04-18 DOI: 10.1112/mtk.12249
Ernie Croot, Hamed Mousavi, Maxie Schmidt

Let be pairwise distinct primes. From a theorem of Kummer, each prime can divide at most times. We show that, for all , if are sufficiently large in terms of and , then there exist infinitely many positive integers such that each divides at most times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers such that is coprime to 105.

设是一对不同的质数。根据库默尔的定理,每个素数最多可以整除多次。我们证明,对于所有 ,如果 和 都足够大,那么存在无穷多个正整数,使得每个正整数都能最多次整除。我们将这一结果与格雷厄姆的一个著名猜想联系起来,即是否存在无限多的整数,使得 与 105 共素。
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引用次数: 0
Shifted convolution sums for averaged over weighted sets 加权平均集的移位卷积和
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-03-29 DOI: 10.1112/mtk.12247
Wing Hong Leung

Let be the Fourier coefficients of an Hecke–Maass cusp form and be those of an Hecke holomorphic or Hecke–Maass cusp form . Let and be a sequence. We show that if for some ,

设 为 Hecke-Maass cusp 形式的傅立叶系数,为 Hecke 全形或 Hecke-Maass cusp 形式的傅立叶系数。设 和 是一个序列。我们证明,如果对于某个 、
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引用次数: 0
Riesz energy, discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus 球面和平面环面上行列式点过程的里兹能、差异和最佳传输
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-03-28 DOI: 10.1112/mtk.12245
Bence Borda, Peter Grabner, Ryan W. Matzke

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere and the flat torus , and the so-called spherical ensemble on , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega-Cerdà on the Riesz -energy of the harmonic ensemble to the nonsingular regime , and as a corollary find the expected value of the spherical cap discrepancy via the Stolarsky invariance principle. We find the expected value of the discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on . We also show that the spherical ensemble and the harmonic ensemble on and with points attain the optimal rate in expectation in the Wasserstein metric , in contrast to independent and identically distributed random points, which are known to lose a factor of .

确定性点过程表现出固有的排斥行为,从而提供了流形上非常均匀分布的点集的例子。在本文中,我们研究了以球面和平面环面上的拉普拉斯特征函数定义的所谓谐波集合,以及源于随机矩阵理论的所谓球面集合。我们将贝尔特兰、马索和奥尔特加-塞尔达关于谐波集合的里兹能的结果扩展到了非奇异制度,并作为推论通过斯托拉斯基不变性原理找到了球顶差异的期望值。我们还找到了调和集合在......上与轴平行的盒和欧几里得球的差异的期望值。我们还证明了球面集合和调和集合上的和点在瓦瑟斯坦度量中达到了期望中的最优率,这与独立且同分布的随机点相反,众所周知,独立且同分布的随机点会损失......倍。
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引用次数: 0
On the entropy and information of Gaussian mixtures 论高斯混合物的熵和信息
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-03-28 DOI: 10.1112/mtk.12246
Alexandros Eskenazis, Lampros Gavalakis

We establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if are independent scalar Gaussian mixtures, then the entropy of is concave in , thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in . As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

我们为居中高斯分布混合物的熵和费雪信息建立了几个凸性质。首先,我们证明,如果是独立标量高斯混合物,那么其熵在 , , 是凹的,从而证实了 Ball、Nayar 和 Tkocz(2016 年)对这类随机变量的猜想。事实上,我们证明了这一论断的广义化,这也加强了 Eskenazis、Nayar 和 Tkocz(2018)的一个结果。对于费雪信息,我们扩展了 Bobkov(2022 年)的一个凸性结果,证明费雪信息矩阵作为作用于......中混合物密度的矩阵值函数,是算子凸性的。作为应用,我们在温和矩假设下,根据中心极限定理建立了加权 i.i.d. 高斯混合物之和的 Fisher 信息矩阵在算子规范中的收敛率。
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引用次数: 0
Moments of zeta and correlations of divisor-sums: Stratification and Vandermonde integrals zeta矩与除数和的相关性:分层和范德蒙德积分
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-03-15 DOI: 10.1112/mtk.12243
Siegfred Baluyot, Brian Conrey

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

我们改进了基廷和第二作者最近开发的启发式。我们的改进为黎曼zeta函数移项矩的猜想渐近公式带来了新的积分表达式。这个表达式类似于布拉德-罗杰斯(Brad Rodgers)和坎南-桑德拉拉詹(Kannan Soundararajan)最近发现的单元群随机矩阵特征多项式矩的公式。
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引用次数: 0
Zeros of modular forms and Faber polynomials 模形式和法布尔多项式的零点
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-03-13 DOI: 10.1112/mtk.12244
Zeév Rudnick

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near vertical lines, with the zeros of a weight form lying at height approximately . This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree .

我们研究了模数群大权重尖顶形式的零点,这些形式在无穷远处有非常大的消失阶数,因此它们在基域有固定数量的有限零点。我们的研究表明,对于大权重形式,这些形式的零点聚集在垂直线附近,一个权重形式的零点位于高度约为 。这与之前已知的情况不同,例如爱森斯坦级数,其零点位于基域边界的圆周部分,或尖顶赫克特征形式的情况,其零点均匀分布在基域中。我们的方法使用法布尔多项式。我们证明,对于我们这一类的尖顶形式,相关的法布尔多项式经过适当的重规范化后,收敛于......度的截断指数多项式。
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引用次数: 0
High moments of theta functions and character sums Theta 函数的高矩和特征和
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-02-14 DOI: 10.1112/mtk.12242
Barnabás Szabó

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet L-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen theorems of Munsch, who proved almost sharp upper bounds for these quantities. The main new ingredient of our proof comes from a paper of Harper, who showed the related result for all under the Riemann Hypothesis. Finally, we obtain a sharp conditional upper bound on high moments of character sums of arbitrary length.

假设存在广义黎曼假设,我们证明了移位狄利克特 L 函数矩的尖锐上界。我们利用它得到了 Theta 函数高矩数的条件上界。这两个结果都加强了芒施的定理,芒施证明了这些量的近乎尖锐的上界。我们证明的主要新成分来自哈珀的一篇论文,他证明了黎曼假设下的所有相关结果。最后,我们得到了任意长度特征和的高矩数的尖锐条件上界。
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引用次数: 0
Hausdorff dimension of Besicovitch sets of Cantor graphs 康托尔图的贝西科维奇集的豪斯多夫维度
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-02-08 DOI: 10.1112/mtk.12241
Iqra Altaf, Marianna Csörnyei, Kornélia Héra

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets Γ, that is, sets that contain a rotated copy of Γ in each direction. We show that for a large class of Cantor sets C and Cantor-graphs Γ built on C, the Hausdorff dimension of any Γ-Besicovitch set must be at least , where .

我们考虑的是可整型集合 Γ 的平面贝西科维奇集合的豪斯多夫维度,即在每个方向上都包含 Γ 的旋转副本的集合。我们证明,对于一大类 Cantor 集 C 和建立在 C 上的 Cantor 图 Γ,任何 Γ-Besicovitch 集的 Hausdorff 维度必须至少为 ,其中 。
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引用次数: 0
On the Lindelöf hypothesis for general sequences 关于一般序列的林德洛夫假设
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-02-05 DOI: 10.1112/mtk.12240
Frederik Broucke, Sebastian Weishäupl

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

在最近的一篇论文中,戈内克、格雷厄姆和李提出了一个关于一般序列的林德洛夫假设(LH)概念,它与正整数序列情况下黎曼zeta函数的通常 LH 相吻合。他们提出了两个猜想:林德洛夫假设应适用于每一个可容许的正整数序列;林德洛夫假设应适用于 "一般 "可容许的正实数序列。在本文中,我们给出了第一个猜想的反例,并证明了第二个猜想可真可假,这取决于 "通用 "的含义:我们构造了产生满足 LH 的序列的概率为 1 的概率过程,并构造了满足 LH 的序列的子空间很小的序列的拜尔拓扑空间。我们还将戈内克、格雷厄姆和李的主要结果,即黎曼假设等价于素数序列的 LH,扩展到贝林广义数系统的背景中。
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引用次数: 0
Zeros of dirichlet L-functions near the critical line 临界线附近的 L 函数的零点
IF 0.8 3区 数学 Q2 Mathematics Pub Date : 2024-01-04 DOI: 10.1112/mtk.12239
George Dickinson

We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet L-functions of modulus q at height T. To do this, we derive an asymptotic for the twisted second moment of Dirichlet L-functions uniformly in q and t. As a second application of the asymptotic formula, we prove that, for every integer q, at least 38.2% of zeros of the primitive Dirichlet L-functions of modulus q lie on the critical line.

作为渐近公式的第二个应用,我们证明了对于每一个整数 q,至少有 38.2% 的模为 q 的基元 Dirichlet L 函数的零点位于临界线上。
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引用次数: 0
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Mathematika
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