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Large deviations of the argument of the Riemann zeta function 黎曼zeta函数参数的大偏差
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-07 DOI: 10.1112/mtk.12251
Alexander Dobner
{"title":"Large deviations of the argument of the Riemann zeta function","authors":"Alexander Dobner","doi":"10.1112/mtk.12251","DOIUrl":"https://doi.org/10.1112/mtk.12251","url":null,"abstract":"<p>Let <span></span><math></math>. We prove an unconditional lower bound on the measure of the sets <span></span><math></math> for <span></span><math></math>. For <span></span><math></math>, our bound has a Gaussian shape with variance proportional to <span></span><math></math>. At the endpoint, <span></span><math></math>, our result implies the best known <span></span><math></math>-theorem for <span></span><math></math> that is due to Tsang. We also explain how the method breaks down for <span></span><math></math> given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12251","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140844649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Markoff–Lagrange spectrum of one-sided shifts 单边移动的马尔科夫-拉格朗日谱
IF 0.8 3区 数学
Mathematika Pub Date : 2024-05-07 DOI: 10.1112/mtk.12250
Hajime Kaneko, Wolfgang Steiner
{"title":"Markoff–Lagrange spectrum of one-sided shifts","authors":"Hajime Kaneko,&nbsp;Wolfgang Steiner","doi":"10.1112/mtk.12250","DOIUrl":"https://doi.org/10.1112/mtk.12250","url":null,"abstract":"<p>For the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are <span></span><math></math>-adic sequences, where <span></span><math></math> is a certain infinite set of substitutions that contains Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff–Lagrange spectrum.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12250","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140844614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mixed-norm estimates via the helicoidal method 通过螺旋线方法进行混合正态估计
IF 0.8 3区 数学
Mathematika Pub Date : 2024-04-21 DOI: 10.1112/mtk.12248
Cristina Benea, Camil Muscalu
{"title":"Mixed-norm estimates via the helicoidal method","authors":"Cristina Benea,&nbsp;Camil Muscalu","doi":"10.1112/mtk.12248","DOIUrl":"https://doi.org/10.1112/mtk.12248","url":null,"abstract":"<p>We prove multiple vector-valued and mixed-norm estimates for multilinear operators in <span></span><math></math>, more precisely for multilinear operators <span></span><math></math> associated to a symbol singular along a <span></span><math></math>-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension <span></span><math></math>, the input functions are not necessarily in <span></span><math></math> and can instead be elements of mixed-norm spaces <span></span><math></math>.</p><p>Such a result has interesting consequences especially when <span></span><math></math> spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy “purely mixed-norm estimates” and no classical <span></span><math></math> estimates.</p><p>Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12248","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140622646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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
Mathematika Pub Date : 2024-04-18 DOI: 10.1112/mtk.12249
Ernie Croot, Hamed Mousavi, Maxie Schmidt
{"title":"On a conjecture of Graham on the -divisibility of central binomial coefficients","authors":"Ernie Croot,&nbsp;Hamed Mousavi,&nbsp;Maxie Schmidt","doi":"10.1112/mtk.12249","DOIUrl":"https://doi.org/10.1112/mtk.12249","url":null,"abstract":"<p>Let <span></span><math></math> be pairwise distinct primes. From a theorem of Kummer, each prime <span></span><math></math> can divide <span></span><math></math> at most <span></span><math></math> times. We show that, for all <span></span><math></math>, if <span></span><math></math> are sufficiently large in terms of <span></span><math></math> and <span></span><math></math>, then there exist infinitely many positive integers <span></span><math></math> such that each <span></span><math></math> divides <span></span><math></math> at most <span></span><math></math> times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers <span></span><math></math> such that <span></span><math></math> is coprime to 105.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12249","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140619726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Mathematika Pub Date : 2024-03-29 DOI: 10.1112/mtk.12247
Wing Hong Leung
{"title":"Shifted convolution sums for averaged over weighted sets","authors":"Wing Hong Leung","doi":"10.1112/mtk.12247","DOIUrl":"https://doi.org/10.1112/mtk.12247","url":null,"abstract":"<p>Let <span></span><math></math> be the Fourier coefficients of an <span></span><math></math> Hecke–Maass cusp form <span></span><math></math> and <span></span><math></math> be those of an <span></span><math></math> Hecke holomorphic or Hecke–Maass cusp form <span></span><math></math>. Let <span></span><math></math> and <span></span><math></math> be a sequence. We show that if <span></span><math></math> for some <span></span><math></math>,\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12247","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140321771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Mathematika Pub Date : 2024-03-28 DOI: 10.1112/mtk.12245
Bence Borda, Peter Grabner, Ryan W. Matzke
{"title":"Riesz energy, discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus","authors":"Bence Borda,&nbsp;Peter Grabner,&nbsp;Ryan W. Matzke","doi":"10.1112/mtk.12245","DOIUrl":"https://doi.org/10.1112/mtk.12245","url":null,"abstract":"<p>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 <span></span><math></math> and the flat torus <span></span><math></math>, and the so-called spherical ensemble on <span></span><math></math>, which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega-Cerdà on the Riesz <span></span><math></math>-energy of the harmonic ensemble to the nonsingular regime <span></span><math></math>, and as a corollary find the expected value of the spherical cap <span></span><math></math> discrepancy via the Stolarsky invariance principle. We find the expected value of the <span></span><math></math> discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on <span></span><math></math>. We also show that the spherical ensemble and the harmonic ensemble on <span></span><math></math> and <span></span><math></math> with <span></span><math></math> points attain the optimal rate <span></span><math></math> in expectation in the Wasserstein metric <span></span><math></math>, in contrast to independent and identically distributed random points, which are known to lose a factor of <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12245","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140310329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Mathematika Pub Date : 2024-03-28 DOI: 10.1112/mtk.12246
Alexandros Eskenazis, Lampros Gavalakis
{"title":"On the entropy and information of Gaussian mixtures","authors":"Alexandros Eskenazis,&nbsp;Lampros Gavalakis","doi":"10.1112/mtk.12246","DOIUrl":"https://doi.org/10.1112/mtk.12246","url":null,"abstract":"<p>We establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if <span></span><math></math> are independent scalar Gaussian mixtures, then the entropy of <span></span><math></math> is concave in <span></span><math></math>, 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 <span></span><math></math>. 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.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12246","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140310328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Mathematika Pub Date : 2024-03-15 DOI: 10.1112/mtk.12243
Siegfred Baluyot, Brian Conrey
{"title":"Moments of zeta and correlations of divisor-sums: Stratification and Vandermonde integrals","authors":"Siegfred Baluyot,&nbsp;Brian Conrey","doi":"10.1112/mtk.12243","DOIUrl":"https://doi.org/10.1112/mtk.12243","url":null,"abstract":"<p>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.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140139250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Mathematika Pub Date : 2024-03-13 DOI: 10.1112/mtk.12244
Zeév Rudnick
{"title":"Zeros of modular forms and Faber polynomials","authors":"Zeév Rudnick","doi":"10.1112/mtk.12244","DOIUrl":"https://doi.org/10.1112/mtk.12244","url":null,"abstract":"<p>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 <math></math> of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near <math></math> vertical lines, with the zeros of a weight <math></math> form lying at height approximately <math></math>. 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 <math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12244","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140114148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Mathematika Pub Date : 2024-02-14 DOI: 10.1112/mtk.12242
Barnabás Szabó
{"title":"High moments of theta functions and character sums","authors":"Barnabás Szabó","doi":"10.1112/mtk.12242","DOIUrl":"https://doi.org/10.1112/mtk.12242","url":null,"abstract":"<p>Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet <i>L</i>-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 <math></math> for all <math></math> under the Riemann Hypothesis. Finally, we obtain a sharp conditional upper bound on high moments of character sums of arbitrary length.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12242","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139739178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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