Ramanujan Journal最新文献

英文中文

A positivity conjecture for a quotient of q-binomial coefficients. 一个关于q-二项式系数商的正性猜想。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2026-01-01 Epub Date: 2025-12-17 DOI: 10.1007/s11139-025-01285-2

M Gatzweiler, C Krattenthaler

We conjecture that, if the quotient of two q-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D. Stanton's fake Gaussian sequences. As a corollary of one of our results we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S. Hopkins and M. Rubey, Selecta Math. (N.S.) 28 (2022), Paper No. 10] is indeed a polynomial with non-negative integer coefficients.

我们推测，如果具有相同顶参数的两个q-二项式系数的商是一个多项式，则它具有非负系数。我们总结了关于这个猜想的已知情况，并用两个非平凡的例子证明了它。此外，我们继续将我们的猜想扩展到D. Stanton的假高斯序列。作为我们其中一个结果的推论，我们得到了一个被推测为Kreweras词的循环筛选多项式的多项式[S]。霍普金斯和鲁比先生，选择数学。(n。)28(2022)，第10号论文]确实是一个系数非负整数的多项式。

引用次数: 0

Computer-assisted construction of Ramanujan-Sato series for 1 over π. 1 / π的Ramanujan-Sato级数的计算机辅助构造。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2026-01-01 Epub Date: 2026-02-26 DOI: 10.1007/s11139-026-01352-2

Ralf Hemmecke, Peter Paule, Cristian-Silviu Radu

Referring to ideas of Sato and Yang in (Math Z 246:1-19, 2004) described a construction of series for 1 over $π$ starting with a pair (g, h), where g is a modular form of weight 2 and h is a modular function; i.e., a modular form of weight zero. In this article we present an algorithmic version, called "Sato construction". Series for $1 / π$ obtained this way will be called "Ramanujan-Sato" series. Famous series fit into this definition, for instance, Ramanujan's series used by Gosper and the series used by the Chudnovsky brothers for computing millions of digits of $π$ . We show that these series are induced by members of infinite families of Sato triples $(N, γ_{N}, τ_{N})$ where $N > 1$ is an integer and $γ_{N}$ a $2 \times 2$ matrix satisfying $γ_{N} τ_{N} = N τ_{N}$ for $τ_{N}$ being an element from the upper half of the complex plane. In addition to procedures for guessing and proving from the holonomic toolbox together with the algorithm "ModFormDE", as described in Paule and Radu in Int J Number Theory (17:713-759, 2021), a central role is played by the algorithm "MultiSamba", an extension of Samba ("subalgebra module basis algorithm") originating from Radu in (J Symb Comput 68:225-253, 2015) and Hemmecke in (J Symb Comput 84:14-24, 2018). With the help of MultiSamba one can find and prove evaluations of modular functions, at imaginary quadratic points, in terms of nested algebraic expressions. As a consequence, all the series for $1 / π$ constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.

参考Sato和Yang在（Math Z 246:1- 19,2004）中的思想，描述了从一对（g, h）开始的1 / π级数的构造，其中g是权值2的模形式，h是模函数；即，权值为零的模形式。在本文中，我们提出了一个算法版本，称为“佐藤构造”。用这种方法得到的1 / π级数称为“Ramanujan-Sato”级数。著名的级数符合这个定义，例如，高斯帕尔使用的拉马努金级数和丘德诺夫斯基兄弟用于计算π的数百万位数字的级数。我们证明了这些级数是由无穷佐藤三元组（N, γ N, τ N）族的成员导出的，其中N > 1是一个整数，γ N是一个2 × 2矩阵，满足γ N τ N = N τ N，其中τ N是复平面上半部分的一个元素。除了从完整工具箱中进行猜测和证明的过程以及算法“ModFormDE”，如Paule和Radu在Int J数论（17:713-759,2021）中所描述的那样，算法“MultiSamba”发挥了核心作用，它是Samba（“子代数模块基算法”）的扩展，源自Radu In （J Symb Comput 68:225- 253,2015）和Hemmecke In （J Symb Comput 84:14- 24,2018）。在MultiSamba的帮助下，可以根据嵌套代数表达式找到并证明虚二次点上模函数的求值。因此，在MultiSamba的帮助下构造的1 / π的所有级数都以严格的非数值方式被完全证明。

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Computer-assisted construction of Ramanujan-Sato series for 1 over <ns0:math><ns0:mi>π</ns0:mi></ns0:math>.","authors":"Ralf Hemmecke, Peter Paule, Cristian-Silviu Radu","doi":"10.1007/s11139-026-01352-2","DOIUrl":"https://doi.org/10.1007/s11139-026-01352-2","url":null,"abstract":"Referring to ideas of Sato and Yang in (Math Z 246:1-19, 2004) described a construction of series for 1 over <math><mi>π</mi></math> starting with a pair (g, h), where g is a modular form of weight 2 and h is a modular function; i.e., a modular form of weight zero. In this article we present an algorithmic version, called \"Sato construction\". Series for <math><mrow><mn>1</mn> <mo>/</mo> <mi>π</mi></mrow> </math> obtained this way will be called \"Ramanujan-Sato\" series. Famous series fit into this definition, for instance, Ramanujan's series used by Gosper and the series used by the Chudnovsky brothers for computing millions of digits of <math><mi>π</mi></math> . We show that these series are induced by members of infinite families of Sato triples <math><mrow><mo>(</mo> <mi>N</mi> <mo>,</mo> <msub><mi>γ</mi> <mi>N</mi></msub> <mo>,</mo> <msub><mi>τ</mi> <mi>N</mi></msub> <mo>)</mo></mrow> </math> where <math><mrow><mi>N</mi> <mo>></mo> <mn>1</mn></mrow> </math> is an integer and <math><msub><mi>γ</mi> <mi>N</mi></msub> </math> a <math><mrow><mn>2</mn> <mo>×</mo> <mn>2</mn></mrow> </math> matrix satisfying <math> <mrow><msub><mi>γ</mi> <mi>N</mi></msub> <msub><mi>τ</mi> <mi>N</mi></msub> <mo>=</mo> <mi>N</mi> <msub><mi>τ</mi> <mi>N</mi></msub> </mrow> </math> for <math><msub><mi>τ</mi> <mi>N</mi></msub> </math> being an element from the upper half of the complex plane. In addition to procedures for guessing and proving from the holonomic toolbox together with the algorithm \"ModFormDE\", as described in Paule and Radu in Int J Number Theory (17:713-759, 2021), a central role is played by the algorithm \"MultiSamba\", an extension of Samba (\"subalgebra module basis algorithm\") originating from Radu in (J Symb Comput 68:225-253, 2015) and Hemmecke in (J Symb Comput 84:14-24, 2018). With the help of MultiSamba one can find and prove evaluations of modular functions, at imaginary quadratic points, in terms of nested algebraic expressions. As a consequence, all the series for <math><mrow><mn>1</mn> <mo>/</mo> <mi>π</mi></mrow> </math> constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"69 3","pages":"73"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12946297/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147328124","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

Arithmetic properties of character degrees and the generalised knutson index. 特征度的算术性质与广义knutson索引。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2026-01-01 Epub Date: 2026-01-23 DOI: 10.1007/s11139-025-01311-3

Diego Martín Duro

In this paper, we introduce the generalised Knutson Index and compute it for the special linear groups and projective special linear groups of degree two by computing the lowest common multiple of the degrees of their irreducible representations. We also classify all alternating and symmetric groups such that the lowest common multiple of the degrees of their irreducible representations equals the order groups, which yields a lower bound on the generalised Knutson Indices of these groups.

本文引入广义Knutson指数，并通过计算二阶特殊线性群和投影线性群的不可约表示的最小公倍数来计算其广义Knutson指数。我们还对所有交替对称群进行了分类，使其不可约表示度的最小公倍数等于序群，从而得到了这些群的广义Knutson指标的下界。

引用次数: 0

A remarkable basic hypergeometric identity. 一个了不起的超几何恒等式。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-01-31 DOI: 10.1007/s11139-024-00994-4

Christian Krattenthaler, Wadim Zudilin

We give a closed form for quotients of truncated basic hypergeometric series where the base q is evaluated at roots of unity.

给出了截断基超几何级数的商的一个封闭形式，其中基q在单位根处求值。

引用次数: 0

Flipping operators and locally harmonic Maass forms. 翻转算子和局部调和质量形式。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-08-08 DOI: 10.1007/s11139-025-01183-7

Kathrin Bringmann, Andreas Mono, Larry Rolen

In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into two parts, and each operator controls one of these parts. A third operator, called the flipping operator, exchanges the role of these two parts. Maass-Poincaré series (of parabolic type) form a convenient basis of negative weight harmonic Maass forms of manageable growth, and flipping has the effect of negating an index. Recently, there has been much interest in locally harmonic Maass forms defined by the first author, Kane, and Kohnen. These are lifts of Poincaré series of hyperbolic type, and are intimately related to the Shimura and Shintani lifts. In this note, we prove that a similar property holds for the flipping operator applied to these Poincaré series.

在可管理增长的积分权调和质量形式理论中，两个关键的微分算子，Bol算子和阴影算子，起着基本的作用。调和质量形式的可管理增长通常分为两个部分，每个算子控制其中一个部分。第三个操作符，称为翻转操作符，交换这两个部分的角色。抛物型的Maass- poincar级数形成了可管理增长的负权和质量形式的便利基础，翻转具有负指标的作用。最近，人们对第一作者Kane和Kohnen定义的局部谐波质量形式很感兴趣。这些升降机属于庞加莱系列双曲型升降机，与志村升降机和新谷升降机密切相关。在这篇笔记中，我们证明了一个类似的性质适用于这些庞卡罗级数的翻转算子。

引用次数: 0

Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin-Lehner involution. 通过Atkin-Lehner对合解释PEND和POND分区之间不可预见的同余关系。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-05-23 DOI: 10.1007/s11139-025-01111-9

James A Sellers, Nicolas Allen Smoot

For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary q-series identities and generating function manipulations, along with mathematical induction: For all $α \geq 1$ and all $n \geq 0,$ $\begin{matrix} pend (3^{2 α + 1} n + \frac{17 \cdot 3^{2 α} - 1}{8}) & \equiv 0 (mod 3), and \\ pond (3^{2 α + 1} n + \frac{23 \cdot 3^{2 α} + 1}{8}) & \equiv 0 (mod 3) \end{matrix}$ where $pend (n)$ counts the number of PEND partitions of weight n and $pond (n)$ counts the number of POND partitions of weight n. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin-Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.

在过去的几年中，许多作者从不同的角度研究了POD和PED分区。这些是整数分区，其中奇数部分必须是不同的（在POD分区的情况下），或者偶数部分必须是不同的（在PED分区的情况下）。最近，Ballantine和Welch被引导考虑POND和PEND分区，它们是整数分区，其中奇数部分不能区分（在POND分区的情况下）或偶数部分不能区分（在PEND分区的情况下）。不久后，第一作者通过初等q级数恒等式和生成函数的操作，结合数学归纳法证明了以下结果：对于所有α≥1,所有n≥0 , 挂件3 2α+ 1 n + 17·3 2α- 1 8≡0 (mod 3),和池塘3 2α+ 1 n + 23·3 2α+ 1 8≡0 3 (mod)挂件(n)计数挂件分区的数量重量n和池塘(n)计数池分区的数量重量n。在这个工作中,我们重新审视这些家庭的刻画,我们通过Atkin-Lehner对合显示它们之间的关系。从这个关系中，我们可以证明，一旦上述同余族中的一个已知，另一个就会紧随其后。

{"title":"Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin-Lehner involution.","authors":"James A Sellers, Nicolas Allen Smoot","doi":"10.1007/s11139-025-01111-9","DOIUrl":"10.1007/s11139-025-01111-9","url":null,"abstract":"For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary q-series identities and generating function manipulations, along with mathematical induction: For all <math><mrow><mi>α</mi> <mo>≥</mo> <mn>1</mn></mrow> </math> and all <math><mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo></mrow> </math> <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><mrow><mspace></mspace> <mtext>pend</mtext> <mspace></mspace></mrow> <mfenced><msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn></mrow> </msup> <mi>n</mi> <mo>+</mo> <mfrac><mrow><mn>17</mn> <mo>·</mo> <msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi></mrow> </msup> <mo>-</mo> <mn>1</mn></mrow> <mn>8</mn></mfrac> </mfenced> </mrow> </mtd> <mtd><mrow><mo>≡</mo> <mn>0</mn> <mspace></mspace> <mo>(</mo> <mo>mod</mo> <mspace></mspace> <mn>3</mn> <mo>)</mo> <mo>,</mo> <mspace></mspace> <mtext>and</mtext></mrow> </mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mrow><mspace></mspace> <mtext>pond</mtext> <mspace></mspace></mrow> <mfenced><msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn></mrow> </msup> <mi>n</mi> <mo>+</mo> <mfrac><mrow><mn>23</mn> <mo>·</mo> <msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi></mrow> </msup> <mo>+</mo> <mn>1</mn></mrow> <mn>8</mn></mfrac> </mfenced> </mrow> </mtd> <mtd><mrow><mo>≡</mo> <mn>0</mn> <mspace></mspace> <mo>(</mo> <mo>mod</mo> <mspace></mspace> <mn>3</mn> <mo>)</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <math> <mrow><mrow><mspace></mspace> <mtext>pend</mtext> <mspace></mspace></mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> counts the number of PEND partitions of weight n and <math> <mrow><mrow><mspace></mspace> <mtext>pond</mtext> <mspace></mspace></mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> counts the number of POND partitions of weight n. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin-Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"67 3","pages":"60"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12102003/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144144343","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

Kloosterman sums on orthogonal groups. 正交群上的Kloosterman和。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-06-23 DOI: 10.1007/s11139-025-01135-1

Catinca Mujdei

We study Kloosterman sums on the orthogonal groups $S O_{3, 3}$ and $S O_{4, 2}$ , associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and p-adic analysis.

研究了与各自Weyl群的短元素相关的正交群so3,3和so4,2上的Kloosterman和。用多维指数和的形式给出了这些和的显式描述。这些是由代数几何和p进分析的方法组合而成的。

引用次数: 0

Normal distribution of bad reduction. 不良还原率正态分布。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-05-22 DOI: 10.1007/s11139-025-01108-4

Robert J Lemke Oliver, Daniel Loughran, Ari Shnidman

We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, $100 %$ of curves in these families have, in a precise sense, many such primes.

证明了曲线族中不良半稳约素数的正态分布规律。因此，我们推断，当按高度排序时，这些科中100%的曲线在精确意义上有许多这样的素数。

引用次数: 0

Irrationality and transcendence questions in the 'poor man's adèle ring'. “穷人的ad<e:1>圈”中的非理性与超越问题。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-06-18 DOI: 10.1007/s11139-025-01132-4

Florian Luca, Wadim Zudilin

We discuss arithmetic questions related to the 'poor man's adèle ring' $A$ whose elements are encoded by sequences ${(t_{p})}_{p}$ indexed by prime numbers, with each $t_{p}$ viewed as a residue in $Z / p Z$ . Our main theorem is about the $A$ -transcendence of the element ${(F_{p} (q))}_{p}$ , where $F_{n} (q)$ (Schur's q-Fibonacci numbers) are the (1, 1)-entries of $2 \times 2$ -matrices $\begin{matrix} (\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}) (\begin{matrix} 1 & 1 \\ q^{2} & 0 \end{matrix}) \dots (\begin{matrix} 1 & 1 \\ q^{n - 2} & 0 \end{matrix}) \end{matrix}$ and $q > 1$ is an integer. This result was previously known for $q > 1$ square free under the GRH.

我们讨论了与“穷人ad环”A有关的算术问题，其元素由素数索引的序列（t p） p编码，每个t p被视为Z / p Z中的残数。我们的主要定理是关于元素（F p (q)） p的A -超越，其中F n (q)（舒尔的q-斐波那契数）是2 × 2矩阵1 1 1 1 1 1 1 q 0 1 1 q 2 0⋯1 1 q n - 20的（1,1）项，q > 1是一个整数。这个结果之前在GRH下是已知的qbbb101平方自由度。

{"title":"Irrationality and transcendence questions in the 'poor man's adèle ring'.","authors":"Florian Luca, Wadim Zudilin","doi":"10.1007/s11139-025-01132-4","DOIUrl":"https://doi.org/10.1007/s11139-025-01132-4","url":null,"abstract":"We discuss arithmetic questions related to the 'poor man's adèle ring' <math><mi>A</mi></math> whose elements are encoded by sequences <math> <msub><mrow><mo>(</mo> <msub><mi>t</mi> <mi>p</mi></msub> <mo>)</mo></mrow> <mi>p</mi></msub> </math> indexed by prime numbers, with each <math><msub><mi>t</mi> <mi>p</mi></msub> </math> viewed as a residue in <math><mrow><mi>Z</mi> <mo>/</mo> <mi>p</mi> <mi>Z</mi></mrow> </math> . Our main theorem is about the <math><mi>A</mi></math> -transcendence of the element <math> <msub><mrow><mo>(</mo> <msub><mi>F</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> <mi>p</mi></msub> </math> , where <math> <mrow><msub><mi>F</mi> <mi>n</mi></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> </mrow> </math> (Schur's q-Fibonacci numbers) are the (1, 1)-entries of <math><mrow><mn>2</mn> <mo>×</mo> <mn>2</mn></mrow> </math> -matrices <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mn>1</mn></mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mi>q</mi></mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <msup><mi>q</mi> <mn>2</mn></msup> </mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> <mo>⋯</mo> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <msup><mi>q</mi> <mrow><mi>n</mi> <mo>-</mo> <mn>2</mn></mrow> </msup> </mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> and <math><mrow><mi>q</mi> <mo>></mo> <mn>1</mn></mrow> </math> is an integer. This result was previously known for <math><mrow><mi>q</mi> <mo>></mo> <mn>1</mn></mrow> </math> square free under the GRH.","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"67 4","pages":"88"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12177006/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144477899","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

Ramanujan's partition generating functions modulo ℓ. 拉马努金分割生成函数模为l。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2025-01-01 Epub Date: 2025-11-04 DOI: 10.1007/s11139-025-01241-0

Kathrin Bringmann, William Craig, Ken Ono

For the partition function p(n), Ramanujan proved the striking identities <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow> <mtable> <mtr> <mtd> <mrow><msub><mi>P</mi> <mn>5</mn></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <munder><mo>∑</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </munder> <mi>p</mi> <mrow><mo>(</mo> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo>)</mo></mrow> <msup><mi>q</mi> <mi>n</mi></msup> </mrow> </mtd> <mtd><mrow><mo>=</mo> <mn>5</mn> <munder><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </munder> <mfrac> <msubsup> <mfenced><msup><mi>q</mi> <mn>5</mn></msup> <mo>;</mo> <msup><mi>q</mi> <mn>5</mn></msup> </mfenced> <mrow><mi>∞</mi></mrow> <mn>5</mn></msubsup> <msubsup><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mrow><mi>∞</mi></mrow> <mn>6</mn></msubsup> </mfrac> <mo>,</mo></mrow> </mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <msub><mi>P</mi> <mn>7</mn></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <munder><mo>∑</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </munder> <mi>p</mi> <mrow><mo>(</mo> <mn>7</mn> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mo>)</mo></mrow> <msup><mi>q</mi> <mi>n</mi></msup> </mrow> </mtd> <mtd><mrow><mo>=</mo> <mn>7</mn> <munder><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </munder> <mfrac> <msubsup> <mfenced><msup><mi>q</mi> <mn>7</mn></msup> <mo>;</mo> <msup><mi>q</mi> <mn>7</mn></msup> </mfenced> <mrow><mi>∞</mi></mrow> <mn>3</mn></msubsup> <msubsup><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mrow><mi>∞</mi></mrow> <mn>4</mn></msubsup> </mfrac> <mo>+</mo> <mn>49</mn> <mi>q</mi> <munder><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </munder> <mfrac> <msubsup> <mfenced><msup><mi>q</mi> <mn>7</mn></msup> <mo>;</mo> <msup><mi>q</mi> <mn>7</mn></msup> </mfenced> <mrow><mi>∞</mi></mrow> <mn>7</mn></msubsup> <msubsup><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mrow><mi>∞</mi></mrow> <mn>8</mn></msubsup> </mfrac> <mo>,</mo></mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <math> <mrow> <msub><mrow><mo>(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo></mrow> <mi>∞</mi></msub> <mo>:</mo> <mo>=</mo> <msub><mo>∏</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>1</mn></mrow> </msub> <mrow><mo>(</mo> <mn>1</mn> <mo>-</mo> <msup><mi>q</mi> <mi>n</mi></msup> <mo>)</mo></mrow> <mo>.</mo></mrow> </math> As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes <math><mrow><mi>ℓ</mi> <mo>≥</mo> <mn>5</mn> <mo>,</mo></mrow> </math> closed form expressions of the power series <dispformula> <math> <mrow><msub><mi>P</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>:</mo> <mo>=</mo> <munder><mo>∑</mo> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </munder> <mi>p</mi> <mrow><mo>(</mo> <mi>ℓ</mi> <mi>n</mi> <mo>-</mo> <m

对于配分函数p(n)， Ramanujan证明了显著恒等式p5 (q): =∑n≥0 p(5n + 4) q n = 5∏n≥1 q 5；q 5∞5 (q; q)∞6,p7 (q): =∑n≥0 P (7n + 5) q n = 7∏n≥1 q 7；Q 7∞3 (Q； Q)∞4 + 49 Q∏n≥1 Q 7；Q 7∞7 (Q； Q)∞8，其中（Q； Q）∞:=∏n≥1 （1 - Q n）。由于这些恒等式包含了他著名的模5和模7同余，所以对于≥5的素数，我们很自然地寻求幂级数P (q): =∑n≥0 P (n - δ l) q n （mod l）的封闭形式表达式，其中δ l: = 2 - 1 24。在本文中，我们证明了P r (q)≡c r T r (q) q r；q r∞(mod r)，其中c r∈Z是显式的，T r (q)是SL 2 (Z)上权值为r - 1的特殊Dirichlet级数的分支值的Hecke迹的生成函数。这是拉马努金同余模5、7和11的一个新证明，因为不存在权4、6和10的非平凡尖角形式。

引用次数: 0

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Ramanujan Journal

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀