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The Ramanujan Journal最新文献

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On near orthogonality of certain k-vectors involving generalized Ramanujan sums 论涉及广义拉马努扬和的某些 k 向量的近正交性
The Ramanujan Journal Pub Date : 2024-05-24 DOI: 10.1007/s11139-024-00874-x
Neha Elizabeth Thomas, K. Vishnu Namboothiri
{"title":"On near orthogonality of certain k-vectors involving generalized Ramanujan sums","authors":"Neha Elizabeth Thomas, K. Vishnu Namboothiri","doi":"10.1007/s11139-024-00874-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00874-x","url":null,"abstract":"<p>The near orthgonality of certain <i>k</i>-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (Duke Math J 16(2):85–90, 1949). We also prove that the weighted average <span>(frac{1}{k^{s(r+1)}}sum limits _{j=1}^{k^s}j^rc_k^{(s)}(j))</span> remains positive for all <span>(rge 1)</span>. Further, we give a lower bound for <span>(max limits _{N}left| sum limits _{j=1}^{N^s}c_k^{(s)}(j) right| )</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153891","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 a generalization of 5-dissections of some infinite q-products 论某些无穷 q 积的 5-剖分的一般化
The Ramanujan Journal Pub Date : 2024-05-22 DOI: 10.1007/s11139-024-00872-z
Channabasavayya, Gedela Kavya Keerthana, Ranganatha Dasappa
{"title":"On a generalization of 5-dissections of some infinite q-products","authors":"Channabasavayya, Gedela Kavya Keerthana, Ranganatha Dasappa","doi":"10.1007/s11139-024-00872-z","DOIUrl":"https://doi.org/10.1007/s11139-024-00872-z","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141108710","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
Exact formula for cubic partitions 立方体分区的精确公式
The Ramanujan Journal Pub Date : 2024-05-21 DOI: 10.1007/s11139-024-00871-0
Lukas Mauth
{"title":"Exact formula for cubic partitions","authors":"Lukas Mauth","doi":"10.1007/s11139-024-00871-0","DOIUrl":"https://doi.org/10.1007/s11139-024-00871-0","url":null,"abstract":"<p>We obtain an exact formula for the cubic partition function and prove a conjecture by Banerjee, Paule, Radu and Zeng.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152167","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
Deviation of the rank and crank modulo 11 等级偏差和曲柄调制 11
The Ramanujan Journal Pub Date : 2024-05-21 DOI: 10.1007/s11139-024-00873-y
Nikolay E. Borozenets
{"title":"Deviation of the rank and crank modulo 11","authors":"Nikolay E. Borozenets","doi":"10.1007/s11139-024-00873-y","DOIUrl":"https://doi.org/10.1007/s11139-024-00873-y","url":null,"abstract":"<p>In this paper, we build on the recent results of Frank Garvan and Rishabh Sarma as well as classical results of Bruce Berndt in order to establish the 11-dissection of the deviations of the rank and crank modulo 11. Using our new dissections, we re-derive the results of Garvan, Atkin, Swinnerton-Dyer, Hussain, Ekin and Chern. By developing and exploiting positivity conditions for quotients of theta functions, we will also prove new rank–crank inequalities and make several conjectures, one of which was recently solved by Kathrin Bringmann and Badri Vishal Pandey. For other applications of our methods, in this paper, we will also prove new congruences for rank moments as well as the Andrews’ smallest parts function and Eisenstein series.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152159","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
Generalized Mahler measures of Laurent polynomials 劳伦多项式的广义马勒度量
The Ramanujan Journal Pub Date : 2024-05-21 DOI: 10.1007/s11139-023-00814-1
Subham Roy
{"title":"Generalized Mahler measures of Laurent polynomials","authors":"Subham Roy","doi":"10.1007/s11139-023-00814-1","DOIUrl":"https://doi.org/10.1007/s11139-023-00814-1","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141114667","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
Arithmetic statistics for Galois deformation rings 伽罗瓦变形环的算术统计
The Ramanujan Journal Pub Date : 2024-05-19 DOI: 10.1007/s11139-024-00839-0
Anwesh Ray, Tom Weston
{"title":"Arithmetic statistics for Galois deformation rings","authors":"Anwesh Ray, Tom Weston","doi":"10.1007/s11139-024-00839-0","DOIUrl":"https://doi.org/10.1007/s11139-024-00839-0","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141123341","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 combinatorial proof of q-log-concavity of q-Eulerian numbers q-Eulerian 数的 q-log-concavity 的组合证明
The Ramanujan Journal Pub Date : 2024-05-18 DOI: 10.1007/s11139-024-00841-6
Xinmiao Liu, Jiangxia Hou, Fengxia Liu
{"title":"A combinatorial proof of q-log-concavity of q-Eulerian numbers","authors":"Xinmiao Liu, Jiangxia Hou, Fengxia Liu","doi":"10.1007/s11139-024-00841-6","DOIUrl":"https://doi.org/10.1007/s11139-024-00841-6","url":null,"abstract":"<p>Carlitz established a <i>q</i>-analog of the Eulerian numbers <span>(A_{n,k}(q))</span> and defined the relationship <span>(A_{n,k}(q)=q^{frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q))</span>. In this paper, by using the combinatorial interpretation of <span>(A_{n,k}^{*}(q))</span> and constructing injective maps, we prove that <span>(A_{n,k}^{*}(q))</span> and <span>(A_{n,k}(q))</span> are <i>q</i>-log-concave, that is, all the coefficients of the polynomials <span>(( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) )</span> and <span>((A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q))</span> are nonnegative for <span>(1&lt; k &lt;n)</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060792","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
Prime numbers as values of nested Beatty sequences 作为嵌套比蒂序列值的质数
The Ramanujan Journal Pub Date : 2024-05-16 DOI: 10.1007/s11139-024-00870-1
Yildirim Akbal
{"title":"Prime numbers as values of nested Beatty sequences","authors":"Yildirim Akbal","doi":"10.1007/s11139-024-00870-1","DOIUrl":"https://doi.org/10.1007/s11139-024-00870-1","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140971330","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
The divisibility of the class number of the imaginary quadratic fields $${mathbb {Q}}(sqrt{1-2m^k})$$ 虚二次域的类数可分性 $${mathbb {Q}}(sqrt{1-2m^k})$$
The Ramanujan Journal Pub Date : 2024-05-16 DOI: 10.1007/s11139-024-00860-3
S. Krishnamoorthy, R. Muneeswaran
{"title":"The divisibility of the class number of the imaginary quadratic fields $${mathbb {Q}}(sqrt{1-2m^k})$$","authors":"S. Krishnamoorthy, R. Muneeswaran","doi":"10.1007/s11139-024-00860-3","DOIUrl":"https://doi.org/10.1007/s11139-024-00860-3","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968340","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
Figurate numbers, forms of mixed type, and their representation numbers 数字、混合形式及其表示数
The Ramanujan Journal Pub Date : 2024-05-16 DOI: 10.1007/s11139-024-00868-9
B. Ramakrishnan, Lalit Vaishya
{"title":"Figurate numbers, forms of mixed type, and their representation numbers","authors":"B. Ramakrishnan, Lalit Vaishya","doi":"10.1007/s11139-024-00868-9","DOIUrl":"https://doi.org/10.1007/s11139-024-00868-9","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140970864","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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