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The Ramanujan Journal最新文献

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Extensions of an identity of Chan and Cooper in the spirit of Ramanujan 以拉曼努扬精神扩展陈与库珀的一个特性
The Ramanujan Journal Pub Date : 2024-07-18 DOI: 10.1007/s11139-024-00906-6
Florian Münkel, Lerna Pehlivan, Kenneth S. Williams
{"title":"Extensions of an identity of Chan and Cooper in the spirit of Ramanujan","authors":"Florian Münkel, Lerna Pehlivan, Kenneth S. Williams","doi":"10.1007/s11139-024-00906-6","DOIUrl":"https://doi.org/10.1007/s11139-024-00906-6","url":null,"abstract":"<p>Chan and Cooper proved that if the integers <span>({c(n) (n=0,1,2,ldots )})</span> are given by </p><span>$$begin{aligned} sum _{n=0}^infty c(n)q^n = prod _{n=1}^infty frac{1}{left( 1-q^nright) ^2left( 1-q^{3n}right) ^2}, end{aligned}$$</span><p>then </p><span>$$begin{aligned} sum _{n=0}^infty c(2n+1)q^n = 2 prod _{n=1}^infty frac{left( 1-q^{2n}right) ^4left( 1-q^{6n}right) ^4}{left( 1-q^nright) ^6left( 1-q^{3n}right) ^6}. end{aligned}$$</span><p>We prove many other results of this type and apply them to the determination of congruence properties of the coefficients.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741234","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 lower bound for the discrepancy in a Sato–Tate type measure Sato-Tate 类型测量中的差异下限
The Ramanujan Journal Pub Date : 2024-07-18 DOI: 10.1007/s11139-024-00909-3
Jishu Das
{"title":"A lower bound for the discrepancy in a Sato–Tate type measure","authors":"Jishu Das","doi":"10.1007/s11139-024-00909-3","DOIUrl":"https://doi.org/10.1007/s11139-024-00909-3","url":null,"abstract":"<p>Let <span>(S_k(N))</span> denote the space of cusp forms of even integer weight <i>k</i> and level <i>N</i>. We prove an asymptotic for the Petersson trace formula for <span>(S_k(N))</span> under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato–Tate distribution for levels not divisible by 8. This generalizes a result of Jung and Sardari (Math Ann 378(1–2):513–557, 2020, Theorem 1.6) for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda (Ramanujan J 20(1):81–89, 2009, Theorem 3) for the distribution of eigenvalues <span>(lambda _{p^2}(f))</span> where <i>f</i> is a Hecke eigenform and <i>p</i> is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights <span>(k_n)</span> such that discrepancy in the analogue distribution obtained in Omar and Mazhouda (Ramanujan J 20(1):81–89, 2009) has a lower bound.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741233","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
Diamonds in the Langlands program: a comprehensive review 朗兰兹计划中的钻石：全面回顾
The Ramanujan Journal Pub Date : 2024-07-18 DOI: 10.1007/s11139-024-00899-2
Shanna Dobson
{"title":"Diamonds in the Langlands program: a comprehensive review","authors":"Shanna Dobson","doi":"10.1007/s11139-024-00899-2","DOIUrl":"https://doi.org/10.1007/s11139-024-00899-2","url":null,"abstract":"<p>A comprehensive review of diamonds, in the sense of Scholze, is presented. The diamond formulations of the Fargues-Fontaine curve and <span>(Bun_G)</span> are reviewed. Principal results centered on the diamond formalism in the global Langlands correspondence and the geometrization of the local Langlands correspondence are given. We conclude with a discussion of future geometrizations.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741236","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 finite nonnegative integer sets with identical representation functions 关于具有相同表示函数的有限非负整数集合
The Ramanujan Journal Pub Date : 2024-07-17 DOI: 10.1007/s11139-024-00903-9
Cui-Fang Sun
{"title":"On finite nonnegative integer sets with identical representation functions","authors":"Cui-Fang Sun","doi":"10.1007/s11139-024-00903-9","DOIUrl":"https://doi.org/10.1007/s11139-024-00903-9","url":null,"abstract":"<p>Let <span>(mathbb {N})</span> be the set of all nonnegative integers. For <span>(Ssubseteq mathbb {N})</span> and <span>(nin mathbb {N})</span>, let the representation function <span>(R_{S}(n))</span> denote the number of solutions of the equation <span>(n=s+s')</span> with <span>(s, s'in S)</span> and <span>(s&lt;s')</span>. In this paper, we determine the structure of <span>(C, Dsubseteq mathbb {N})</span> with <span>(Ccup D=[0, m])</span>, <span>(Ccap D={r_{1}, r_{2}})</span>, <span>(r_{1}&lt;r_{2})</span> and <span>(2not mid r_{1})</span> such that <span>(R_{C}(n)=R_{D}(n))</span> for any nonnegative integer <i>n</i>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721598","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
Reconstruction and best approximate inversion formulas for the modified Whittaker–Stockwell transform 改良惠特克-斯托克韦尔变换的重建和最佳近似反演公式
The Ramanujan Journal Pub Date : 2024-07-16 DOI: 10.1007/s11139-024-00900-y
Fethi Soltani
{"title":"Reconstruction and best approximate inversion formulas for the modified Whittaker–Stockwell transform","authors":"Fethi Soltani","doi":"10.1007/s11139-024-00900-y","DOIUrl":"https://doi.org/10.1007/s11139-024-00900-y","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"8 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141642054","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
Rogers–Ramanujan type identities involving double, triple and quadruple sums 涉及二重、三重和四重和的罗杰斯-拉马努扬式等差数列
The Ramanujan Journal Pub Date : 2024-07-12 DOI: 10.1007/s11139-024-00901-x
Zhi Li, Liuquan Wang
{"title":"Rogers–Ramanujan type identities involving double, triple and quadruple sums","authors":"Zhi Li, Liuquan Wang","doi":"10.1007/s11139-024-00901-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00901-x","url":null,"abstract":"<p>We prove a number of new Rogers–Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the literature. This is achieved by direct summation or the constant term method. We also obtain some new single-sum identities as consequences.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141610103","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
Partitions into Segal–Piatetski–Shapiro sequences Segal-Piatetski-Shapiro 序列的分区
The Ramanujan Journal Pub Date : 2024-07-10 DOI: 10.1007/s11139-024-00896-5
Ya-Li Li, Nian Hong Zhou
{"title":"Partitions into Segal–Piatetski–Shapiro sequences","authors":"Ya-Li Li, Nian Hong Zhou","doi":"10.1007/s11139-024-00896-5","DOIUrl":"https://doi.org/10.1007/s11139-024-00896-5","url":null,"abstract":"<p>Let <span>(kappa )</span> be any positive real number and <span>(min mathbb {N}cup {infty })</span> be given. Let <span>(p_{kappa , m}(n))</span> denote the number of partitions of <i>n</i> into the parts from the Segal–Piatestki–Shapiro sequence <span>((lfloor ell ^{kappa }rfloor )_{ell in mathbb {N}})</span> with at most <i>m</i> possible repetitions. In this paper, we establish some asymptotic formulas of Hardy–Ramanujan type for <span>(p_{kappa , m}(n))</span>. As a necessary step in the proof, we prove that the Dirichlet series <span>(zeta _kappa (s)=sum _{nge 1}lfloor n^{kappa }rfloor ^{-s})</span> can be continued analytically beyond the imaginary axis except for simple poles at <span>(s=1/kappa -j, ~(0le j&lt; 1/kappa , jin mathbb {Z}))</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141587753","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 finiteness of solutions for certain Diophantine equations 论某些 Diophantine 方程解的有限性
The Ramanujan Journal Pub Date : 2024-07-10 DOI: 10.1007/s11139-024-00897-4
Mohamed Ouzahra
{"title":"On the finiteness of solutions for certain Diophantine equations","authors":"Mohamed Ouzahra","doi":"10.1007/s11139-024-00897-4","DOIUrl":"https://doi.org/10.1007/s11139-024-00897-4","url":null,"abstract":"<p>We study Diophantine equations of the form <span>( f(n)=m^2 pm a,; n, min mathbb {N}, )</span>where <span>(ain mathbb {N}^*)</span> and <span>(f: mathbb {N} rightarrow mathbb {N})</span> tends to <span>(+infty . )</span> Necessary and sufficient conditions for the set of solutions to be finite are formulated in terms of asymptotic properties and the repartition of the digits of the fractional part of <span>(sqrt{f(n)})</span></p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566686","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 note on two of Ramanujan’s mock theta functions 关于拉曼南的两个模拟 Theta 函数的说明
The Ramanujan Journal Pub Date : 2024-07-09 DOI: 10.1007/s11139-024-00843-4
Richard J. McIntosh
{"title":"A note on two of Ramanujan’s mock theta functions","authors":"Richard J. McIntosh","doi":"10.1007/s11139-024-00843-4","DOIUrl":"https://doi.org/10.1007/s11139-024-00843-4","url":null,"abstract":"<p>In the “Lost” Notebook Ramanujan defined two mock theta functions (which we denote by <span>(phi _{{}_R})</span> and <span>(xi _{{}_R})</span>) and gave relations connecting them to some of the sixth-order mock theta functions. It turns out that these mock theta functions are related to each other by modular transformation. Their transformation formulas are similar to the transformation formulas for the second-order mock theta functions, and involve the same Mordell integrals. We prove several relations involving <span>(phi _{{}_R})</span> and <span>(xi _{{}_R})</span>, and some relations connecting them to some of the second-order mock theta functions. Some alternate formulas for the second-order mock theta function <span>(mu )</span> are given.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566680","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
Bounds on the Möbius-signed partition numbers 莫比乌斯符号分割数的界限
The Ramanujan Journal Pub Date : 2024-07-08 DOI: 10.1007/s11139-024-00885-8
Taylor Daniels
{"title":"Bounds on the Möbius-signed partition numbers","authors":"Taylor Daniels","doi":"10.1007/s11139-024-00885-8","DOIUrl":"https://doi.org/10.1007/s11139-024-00885-8","url":null,"abstract":"<p>For <span>(n in mathbb {N})</span> let <span>(Pi [n])</span> denote the set of partitions of <i>n</i>, i.e., the set of positive integer tuples <span>((x_1,x_2,ldots ,x_k))</span> such that <span>(x_1 ge x_2 ge ldots ge x_k)</span> and <span>(x_1 + x_2 + cdots + x_k = n)</span>. Fixing <span>(f:mathbb {N}rightarrow {0,pm 1})</span>, for <span>(pi = (x_1,x_2,ldots ,x_k) in Pi [n])</span> let <span>(f(pi ) := f(x_1)f(x_2)cdots f(x_k))</span>. In this way we define the signed partition numbers </p><span>$$begin{aligned} p(n,f) = sum _{pi in Pi [n]} f(pi ). end{aligned}$$</span><p>Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for <span>(p(n,mu ))</span> and <span>(p(n,lambda ))</span>, where <span>(mu )</span> and <span>(lambda )</span> denote the Möbius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities <i>p</i>(<i>n</i>, <i>f</i>) generalize the classical notion of restricted partitions.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566818","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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