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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

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Continued fractions and Hardy sums
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-11-21 DOI: 10.1007/s12188-024-00283-3
Alessandro Lägeler
{"title":"Continued fractions and Hardy sums","authors":"Alessandro Lägeler","doi":"10.1007/s12188-024-00283-3","DOIUrl":"10.1007/s12188-024-00283-3","url":null,"abstract":"<div><p>The classical Dedekind sums <i>s</i>(<i>d</i>, <i>c</i>) can be represented as sums over the partial quotients of the continued fraction expansion of the rational <span>(frac{d}{c})</span>. Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of <span>(theta )</span>-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial quotients of these continued fractions. As an application, we prove that the graph of the Hardy sums is dense in <span>(textbf{R}times textbf{Z})</span>.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"107 - 128"},"PeriodicalIF":0.4,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s12188-024-00283-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142821457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Infinite order linear difference equation satisfied by a refinement of Goss zeta function 由 Goss Zeta 函数细化满足的无穷阶线性差分方程
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-11-07 DOI: 10.1007/s12188-024-00284-2
Su Hu, Min-Soo Kim
{"title":"Infinite order linear difference equation satisfied by a refinement of Goss zeta function","authors":"Su Hu,&nbsp;Min-Soo Kim","doi":"10.1007/s12188-024-00284-2","DOIUrl":"10.1007/s12188-024-00284-2","url":null,"abstract":"<div><p>At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function <span>(zeta (s))</span> is not the solution of any algebraic ordinary differential equations on its region of analyticity. Let <i>T</i> be an infinite order linear differential operator introduced by Van Gorder in 2015. Recently, Prado and Klinger-Logan [9] showed that the Hurwitz zeta function <span>(zeta (s,a))</span> formally satisfies the following linear differential equation </p><div><div><span>$$begin{aligned} Tleft[ zeta (s,a) - frac{1}{a^s}right] = frac{1}{(s-1)a^{s-1}}. end{aligned}$$</span></div></div><p>Then in [6], by defining <span>(T_{p}^{a})</span>, a <i>p</i>-adic analogue of Van Gorder’s operator <i>T</i>,  we constructed the following convergent infinite order linear differential equation satisfied by the <i>p</i>-adic Hurwitz-type Euler zeta function <span>(zeta _{p,E}(s,a))</span></p><div><div><span>$$begin{aligned} T_{p}^{a}left[ zeta _{p,E}(s,a)-langle arangle ^{1-s}right] =frac{1}{s-1}left( langle a-1 rangle ^{1-s}-langle arangle ^{1-s}right) . end{aligned}$$</span></div></div><p>In this paper, we consider this problem in the positive characteristic case. That is, by introducing <span>(zeta _{infty }(s_{0},s,a,n))</span>, a Hurwitz type refinement of Goss zeta function, and an infinite order linear difference operator <i>L</i>, we establish the following difference equation </p><div><div><span>$$begin{aligned} Lleft[ zeta _{infty }left( frac{1}{T},s,a,0right) right] =sum _{gamma in mathbb {F}_{q}} frac{1}{langle a+gamma rangle ^{s}}. end{aligned}$$</span></div></div></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"129 - 143"},"PeriodicalIF":0.4,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142821347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Representations of large Mackey Lie algebras and universal tensor categories 大麦基李代数和通用张量范畴的表征
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-09-14 DOI: 10.1007/s12188-024-00280-6
Ivan Penkov, Valdemar Tsanov
{"title":"Representations of large Mackey Lie algebras and universal tensor categories","authors":"Ivan Penkov,&nbsp;Valdemar Tsanov","doi":"10.1007/s12188-024-00280-6","DOIUrl":"10.1007/s12188-024-00280-6","url":null,"abstract":"<div><p>We extend previous work by constructing a universal abelian tensor category <span>(textbf{T}_t)</span> generated by two objects <i>X</i>, <i>Y</i> equipped with finite filtrations <span>(0subsetneq X_0subsetneq ...subsetneq X_{t+1}= X)</span> and <span>(0subsetneq Y_0subsetneq ... subsetneq Y_{t+1}= Y)</span>, and with a pairing <span>(Xotimes Yrightarrow mathbbm {1})</span>, where <span>(mathbbm {1})</span> is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra <span>(mathfrak {gl}^M(V,V_*))</span> of cardinality <span>(2^{aleph _t})</span>, associated to a diagonalizable pairing between two vector spaces <span>(V,V_*)</span> of dimension <span>(aleph _t)</span> over an algebraically closed field <span>({{mathbb {K}}})</span> of characteristic 0. As a preliminary step, we study a tensor category <span>({{mathbb {T}}}_t)</span> generated by the algebraic duals <span>(V^*)</span> and <span>((V_*)^*)</span>. The injective hull of the trivial module <span>({{mathbb {K}}})</span> in <span>({{mathbb {T}}}_t)</span> is a commutative algebra <i>I</i>, and the category <span>(textbf{T}_t)</span> consists of all free <i>I</i>-modules in <span>({{mathbb {T}}}_t)</span>. An essential novelty in our work is the explicit computation of Ext-spaces between simples in both categories <span>(textbf{T}_t)</span> and <span>({{mathbb {T}}}_t)</span>, which had been an open problem already for <span>(t=0)</span>. This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"235 - 278"},"PeriodicalIF":0.4,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s12188-024-00280-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Ramanujan expansions and primes in arithmetic progressions 论算术级数中的拉马努扬展开和素数
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-08-23 DOI: 10.1007/s12188-024-00282-4
Maurizio Laporta
{"title":"On Ramanujan expansions and primes in arithmetic progressions","authors":"Maurizio Laporta","doi":"10.1007/s12188-024-00282-4","DOIUrl":"10.1007/s12188-024-00282-4","url":null,"abstract":"<div><p>A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"209 - 224"},"PeriodicalIF":0.4,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s12188-024-00282-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
A Fourier analysis of quadratic Riemann sums and Local integrals of (varvec{zeta (s)}) 二次黎曼和的傅立叶分析以及 $$varvec{zeta (s)}$$ 的局部积分
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-08-09 DOI: 10.1007/s12188-024-00278-0
Michel J. G. Weber
{"title":"A Fourier analysis of quadratic Riemann sums and Local integrals of (varvec{zeta (s)})","authors":"Michel J. G. Weber","doi":"10.1007/s12188-024-00278-0","DOIUrl":"10.1007/s12188-024-00278-0","url":null,"abstract":"<div><p>Let <span>(zeta (s))</span>, <span>(s={sigma }+it)</span>, be the Riemann zeta function. We use Fourier analysis to obtain, after a preliminary study of quadratic Riemann sums, a precise formula of the local integrals <span>(int _n^{n+1} |zeta ({sigma }+it ) |^2 textrm{d}t)</span>, for <span>(frac{1}{2}&lt;{sigma }&lt;1)</span>. We also study related <span>(mathcal {S}^{2})</span>-Stepanov norms of <span>(zeta (s))</span> in connection with the strong Voronin Universality Theorem.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"145 - 161"},"PeriodicalIF":0.4,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141924007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
The adjoint of the nullwert map on Jacobi forms of lattice index 网格指数雅可比形式上的空值映射的邻接点
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-07-25 DOI: 10.1007/s12188-024-00281-5
Hatice Boylan
{"title":"The adjoint of the nullwert map on Jacobi forms of lattice index","authors":"Hatice Boylan","doi":"10.1007/s12188-024-00281-5","DOIUrl":"10.1007/s12188-024-00281-5","url":null,"abstract":"<div><p>We state and prove a formula for the adjoint of the nullwert map from spaces of Jacobi cusp forms of lattice index to spaces of modular forms. Furthermore, we prove a nonvanishing result for the image of the adjoint of the nullwert map.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"225 - 234"},"PeriodicalIF":0.4,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141774736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms 论从比安奇模态到西格尔模态的 Theta 提升的非凡性
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-07-19 DOI: 10.1007/s12188-024-00279-z
Di Zhang
{"title":"On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms","authors":"Di Zhang","doi":"10.1007/s12188-024-00279-z","DOIUrl":"10.1007/s12188-024-00279-z","url":null,"abstract":"<div><p>In this paper we study the theta lifting of a weight 2 Bianchi modular form <span>({mathcal {F}})</span> of level <span>(Gamma _0({mathfrak {n}}))</span> with <span>({mathfrak {n}})</span> square-free to a weight 2 holomorphic Siegel modular form. Motivated by Prasanna’s work for the Shintani lifting, we define the local Schwartz function at finite places using a quadratic Hecke character <span>(chi )</span> of square-free conductor <span>({mathfrak {f}})</span> coprime to level <span>({mathfrak {n}})</span>. Then, at certain 2 by 2 g matrices <span>(beta )</span> related to <span>({mathfrak {f}})</span>, we can express the Fourier coefficient of this theta lifting as a multiple of <span>(L({mathcal {F}},chi ,1))</span> by a non-zero constant. If the twisted <i>L</i>-value is known to be non-vanishing, we can deduce the non-vanishing of our theta lifting.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 2","pages":"163 - 208"},"PeriodicalIF":0.4,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Connectivity properties of the Schur–Horn map for real Grassmannians 实格拉斯曼人的舒尔-霍恩图谱的连接特性
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-05-06 DOI: 10.1007/s12188-024-00277-1
Augustin-Liviu Mare
{"title":"Connectivity properties of the Schur–Horn map for real Grassmannians","authors":"Augustin-Liviu Mare","doi":"10.1007/s12188-024-00277-1","DOIUrl":"10.1007/s12188-024-00277-1","url":null,"abstract":"<div><p>To any <i>V</i> in the Grassmannian <span>(textrm{Gr}_k({mathbb R}^n))</span> of <i>k</i>-dimensional vector subspaces in <span>({mathbb {R}}^n)</span> one can associate the diagonal entries of the (<span>(ntimes n)</span>) matrix corresponding to the orthogonal projection of <span>({mathbb {R}}^n)</span> to <i>V</i>. One obtains a map <span>(textrm{Gr}_k({mathbb {R}}^n)rightarrow {mathbb {R}}^n)</span> (the Schur–Horn map). The main result of this paper is a criterion for pre-images of vectors in <span>({mathbb {R}}^n)</span> to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill et al. (SIAM J Appl Algebra Geom 1:38–72, 2017).</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 1","pages":"33 - 55"},"PeriodicalIF":0.4,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Lifts of line bundles on curves on K3 surfaces K3 表面曲线上线束的提升
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-04-16 DOI: 10.1007/s12188-024-00275-3
Kenta Watanabe, Jiryo Komeda
{"title":"Lifts of line bundles on curves on K3 surfaces","authors":"Kenta Watanabe,&nbsp;Jiryo Komeda","doi":"10.1007/s12188-024-00275-3","DOIUrl":"10.1007/s12188-024-00275-3","url":null,"abstract":"<div><p>Let <i>X</i> be a K3 surface, let <i>C</i> be a smooth curve of genus <i>g</i> on <i>X</i>, and let <i>A</i> be a line bundle of degree <i>d</i> on <i>C</i>. Then a line bundle <i>M</i> on <i>X</i> with <span>(Motimes {mathcal {O}}_C=A)</span> is called a lift of <i>A</i>. In this paper, we prove that if the dimension of the linear system |<i>A</i>| is <span>(rge 2)</span>, <span>(g&gt;2d-3+(r-1)^2)</span>, <span>(dge 2r+4)</span>, and <i>A</i> computes the Clifford index of <i>C</i>, then there exists a base point free lift <i>M</i> of <i>A</i> such that the general member of |<i>M</i>| is a smooth curve of genus <i>r</i>. In particular, if |<i>A</i>| is a base point free net which defines a double covering <span>(pi :Clongrightarrow C_0)</span> of a smooth curve <span>(C_0subset {mathbb {P}}^2)</span> of degree <span>(kge 4)</span> branched at distinct 6<i>k</i> points on <span>(C_0)</span>, then, by using the aforementioned result, we can also show that there exists a 2:1 morphism <span>({tilde{pi }}:Xlongrightarrow {mathbb {P}}^2)</span> such that <span>({tilde{pi }}|_C=pi )</span>.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 1","pages":"95 - 106"},"PeriodicalIF":0.4,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
The distribution of the multiplicative index of algebraic numbers over residue classes 代数数的乘法指数在残差类上的分布
IF 0.4 4区 数学
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Pub Date : 2024-04-09 DOI: 10.1007/s12188-024-00276-2
Pieter Moree, Antonella Perucca, Pietro Sgobba
{"title":"The distribution of the multiplicative index of algebraic numbers over residue classes","authors":"Pieter Moree,&nbsp;Antonella Perucca,&nbsp;Pietro Sgobba","doi":"10.1007/s12188-024-00276-2","DOIUrl":"10.1007/s12188-024-00276-2","url":null,"abstract":"<div><p>Let <i>K</i> be a number field and <i>G</i> a finitely generated torsion-free subgroup of <span>(K^times )</span>. Given a prime <span>(mathfrak {p})</span> of <i>K</i> we denote by <span>({{,textrm{ind},}}_mathfrak {p}(G))</span> the index of the subgroup <span>((Gbmod mathfrak {p}))</span> of the multiplicative group of the residue field at <span>(mathfrak {p})</span>. Under the Generalized Riemann Hypothesis we determine the natural density of primes of <i>K</i> for which this index is in a prescribed set <i>S</i> and has prescribed Frobenius in a finite Galois extension <i>F</i> of <i>K</i>. We study in detail the natural density in case <i>S</i> is an arithmetic progression, in particular its positivity.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":"94 1","pages":"1 - 17"},"PeriodicalIF":0.4,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140595195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
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