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The Ramanujan Journal最新文献

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Evaluation of Hecke–Rogers series and expansions of the rank function 赫克-罗杰斯数列的评估和秩函数的展开
The Ramanujan Journal Pub Date : 2024-07-05 DOI: 10.1007/s11139-024-00842-5
J. G. Bradley-Thrush
{"title":"Evaluation of Hecke–Rogers series and expansions of the rank function","authors":"J. G. Bradley-Thrush","doi":"10.1007/s11139-024-00842-5","DOIUrl":"https://doi.org/10.1007/s11139-024-00842-5","url":null,"abstract":"<p>A formula is established for the evaluation of double series of Hecke–Rogers type in terms of theta functions and Appell–Lerch functions. This formula is similar to others obtained previously by Hickerson and Mortenson, and by Mortenson and Zwegers. It is applied to the rank function, leading to an expansion closely analogous to two of Garvan’s double series identities. Several identities involving third-order mock theta functions are obtained as special cases.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141577190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Normal integral bases of Lehmer’s cyclic quintic fields 雷默循环五元场的常积分基
The Ramanujan Journal Pub Date : 2024-07-05 DOI: 10.1007/s11139-024-00875-w
Yu Hashimoto, Miho Aoki
{"title":"Normal integral bases of Lehmer’s cyclic quintic fields","authors":"Yu Hashimoto, Miho Aoki","doi":"10.1007/s11139-024-00875-w","DOIUrl":"https://doi.org/10.1007/s11139-024-00875-w","url":null,"abstract":"<p>Let <span>(K_n)</span> be a tamely ramified cyclic quintic field generated by a root of Emma Lehmer’s parametric polynomial. We give all normal integral bases for <span>(K_n)</span> only by the roots of the polynomial, which is a generalization of the work of Lehmer in the case that <span>(n^4+5n^3+15n^2+25n+25)</span> is prime number, and Spearman–Willliams in the case that <span>(n^4+5n^3+15n^2+25n+25)</span> is square free.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Construction of Jacobi forms using adjoint of the Jacobi–Serre derivative 利用雅各比-塞尔导数的邻接构建雅各比形式
The Ramanujan Journal Pub Date : 2024-07-04 DOI: 10.1007/s11139-024-00890-x
Mrityunjoy Charan, Lalit Vaishya
{"title":"Construction of Jacobi forms using adjoint of the Jacobi–Serre derivative","authors":"Mrityunjoy Charan, Lalit Vaishya","doi":"10.1007/s11139-024-00890-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00890-x","url":null,"abstract":"<p>In the article, we study the Oberdieck derivative defined on the space of weak Jacobi forms. We prove that the Oberdieck derivative maps a Jacobi form to a Jacobi form. Moreover, we study the adjoint of the Oberdieck derivative of a Jacobi cusp form with respect to the Petersson scalar product defined on the space of Jacobi forms. As a consequence, we also obtain the adjoint of the Jacobi–Serre derivative (defined in an unpublished work of Oberdieck). As an application, we obtain certain relations among the Fourier coefficients of Jacobi forms.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A vertex operator reformulation of the Kanade–Russell conjecture modulo 9 卡纳德-拉塞尔猜想的顶点算子模9重述
The Ramanujan Journal Pub Date : 2024-07-02 DOI: 10.1007/s11139-024-00895-6
Shunsuke Tsuchioka
{"title":"A vertex operator reformulation of the Kanade–Russell conjecture modulo 9","authors":"Shunsuke Tsuchioka","doi":"10.1007/s11139-024-00895-6","DOIUrl":"https://doi.org/10.1007/s11139-024-00895-6","url":null,"abstract":"<p>We reformulate the Kanade–Russell conjecture modulo 9 via the vertex operators for the level 3 standard modules of type <span>(D^{(3)}_{4})</span>. Along the same lines, we arrive at three partition theorems which may be regarded as an <span>(A^{(2)}_{4})</span> analog of the conjecture. Andrews–van Ekeren–Heluani have proven one of them, and we point out that the others are easily proven from their results.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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 coefficients of automorphic representations over polynomials 论多项式上的自动表征系数
The Ramanujan Journal Pub Date : 2024-07-01 DOI: 10.1007/s11139-024-00889-4
Shu Luo, Huixue Lao
{"title":"On the coefficients of automorphic representations over polynomials","authors":"Shu Luo, Huixue Lao","doi":"10.1007/s11139-024-00889-4","DOIUrl":"https://doi.org/10.1007/s11139-024-00889-4","url":null,"abstract":"<p>Let <span>(pi )</span> be a cuspidal automorphic representation of <span>(textrm{GL}_2(mathbb {A}_mathbb {Q}))</span> associated to holomorphic forms with Fourier coefficients <span>(a_{ pi }(n))</span>. Consider an automorphic representation <span>(Pi )</span> which is equivalent to <span>(textrm{sym}^m pi )</span> or <span>(pi times textrm{sym}^m pi )</span>. We establish uniform upper bounds for <span>(sum _{nleqslant X} |a_{Pi } (|f(n)|)|)</span>, where <span>(f(x)in mathbb {Z}[x])</span> is a polynomial of arbitrary degree. This builds on the work of Chiriac and Yang, and refines one of their results.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Explicit constructions of Diophantine tuples over finite fields 有限域上 Diophantine 元组的显式构造
The Ramanujan Journal Pub Date : 2024-06-29 DOI: 10.1007/s11139-024-00888-5
Seoyoung Kim, Chi Hoi Yip, Semin Yoo
{"title":"Explicit constructions of Diophantine tuples over finite fields","authors":"Seoyoung Kim, Chi Hoi Yip, Semin Yoo","doi":"10.1007/s11139-024-00888-5","DOIUrl":"https://doi.org/10.1007/s11139-024-00888-5","url":null,"abstract":"<p>A Diophantine <i>m</i>-tuple over a finite field <span>({mathbb F}_q)</span> is a set <span>({a_1,ldots , a_m})</span> of <i>m</i> distinct elements in <span>(mathbb {F}_{q}^{*})</span> such that <span>(a_{i}a_{j}+1)</span> is a square in <span>({mathbb F}_q)</span> whenever <span>(ine j)</span>. In this paper, we study <i>M</i>(<i>q</i>), the maximum size of a Diophantine tuple over <span>({mathbb F}_q)</span>, assuming the characteristic of <span>({mathbb F}_q)</span> is fixed and <span>(q rightarrow infty )</span>. By explicit constructions, we improve the lower bound on <i>M</i>(<i>q</i>). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An overpartition analogue of Bressoud’s conjecture for even moduli 偶数模的布列索猜想的过分区类似物
The Ramanujan Journal Pub Date : 2024-06-28 DOI: 10.1007/s11139-024-00887-6
Y. H. Chen, T. T. Gu, T. Y. He, F. Tang, J. J. Wei
{"title":"An overpartition analogue of Bressoud’s conjecture for even moduli","authors":"Y. H. Chen, T. T. Gu, T. Y. He, F. Tang, J. J. Wei","doi":"10.1007/s11139-024-00887-6","DOIUrl":"https://doi.org/10.1007/s11139-024-00887-6","url":null,"abstract":"<p>In 1980, Bressoud conjectured a combinatorial identity <span>(A_j=B_j)</span> for <span>(j=0)</span> or 1. In this paper, we introduce a new partition function <span>(widetilde{B}_0)</span> which can be viewed as an overpartition analogue of the partition function <span>(B_0)</span>. An overpartition is a partition such that the last occurrence of a part can be overlined. We build a bijection to get a relationship between <span>(widetilde{B}_0)</span> and <span>(B_1)</span>, based on which an overpartition analogue of Bressoud’s conjecture for <span>(j=0)</span> is obtained.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sums of logarithmic weights involving r-full numbers 涉及 r 个整数的对数权重之和
The Ramanujan Journal Pub Date : 2024-06-28 DOI: 10.1007/s11139-024-00891-w
Isao Kiuchi
{"title":"Sums of logarithmic weights involving r-full numbers","authors":"Isao Kiuchi","doi":"10.1007/s11139-024-00891-w","DOIUrl":"https://doi.org/10.1007/s11139-024-00891-w","url":null,"abstract":"<p>Let (<i>n</i>, <i>q</i>) denote the greatest common divisor of positive integers <i>n</i> and <i>q</i>, and let <span>(f_{r})</span> denote the characteristic function of <i>r</i>-full numbers. We consider several asymptotic formulas for sums of the modified square-full (<span>(r=2)</span>) and cube-full numbers (<span>(r=3)</span>), which is <span>(sum _{nle y}sum _{qle x}sum _{d|(n,q)}df_{r}left( frac{q}{d}right) log frac{x}{q})</span> with any positive real numbers <i>x</i> and <i>y</i>. Moreover, we derive the asymptotic formula of the above with <span>(r=2)</span> under the Riemann Hypothesis.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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 vanishing coefficients of odd powers of Ramanujan’s theta functions 论拉马努扬 Theta 函数奇次幂的消失系数
The Ramanujan Journal Pub Date : 2024-06-25 DOI: 10.1007/s11139-024-00882-x
Ji-Cai Liu
{"title":"On the vanishing coefficients of odd powers of Ramanujan’s theta functions","authors":"Ji-Cai Liu","doi":"10.1007/s11139-024-00882-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00882-x","url":null,"abstract":"<p>Various vanishing coefficient results on <i>q</i>-series expansions have been widely studied by many authors in recent years. Motivated by these works, we establish a general vanishing coefficient result on odd powers of Ramanujan’s theta functions.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rankin–Cohen brackets of Hilbert Hecke eigenforms 希尔伯特赫克特征形式的兰金-科恩括号
The Ramanujan Journal Pub Date : 2024-06-24 DOI: 10.1007/s11139-024-00883-w
Yichao Zhang, Yang Zhou
{"title":"Rankin–Cohen brackets of Hilbert Hecke eigenforms","authors":"Yichao Zhang, Yang Zhou","doi":"10.1007/s11139-024-00883-w","DOIUrl":"https://doi.org/10.1007/s11139-024-00883-w","url":null,"abstract":"<p>Over any fixed totally real number field with narrow class number one, we prove that the Rankin–Cohen bracket of two Hecke eigenforms for the Hilbert modular group can only be a Hecke eigenform for dimension reasons, except for a couple of cases where the Rankin–Selberg method does not apply. We shall also prove a conjecture of Freitag on the volume of Hilbert modular groups, and assuming a conjecture of Freitag on the dimension of the cuspform space, we obtain a finiteness result on eigenform product identities.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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