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Generalizations of Rogers–Ramanujan type identities 罗杰斯-拉马努扬类等式的一般化
The Ramanujan Journal Pub Date : 2024-08-03 DOI: 10.1007/s11139-024-00918-2
Li-Jun Hao, Xueya Kuai, Lan Xia
{"title":"Generalizations of Rogers–Ramanujan type identities","authors":"Li-Jun Hao, Xueya Kuai, Lan Xia","doi":"10.1007/s11139-024-00918-2","DOIUrl":"https://doi.org/10.1007/s11139-024-00918-2","url":null,"abstract":"<p>Recently the integral method was widely used to prove some Nahm problems. In the present paper we apply this method and the three-term transformation formula for <span>({}_2phi _1)</span> series to establish some multi-sum Rogers-Ramanujan type identities with parameters. As special cases, we derive known Rogers-Ramanujan type identities, also find some new identities.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941477","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
Chebyshev’s bias for Fermat curves of prime degree 素度费马曲线的切比雪夫偏差
The Ramanujan Journal Pub Date : 2024-08-02 DOI: 10.1007/s11139-024-00913-7
Yoshiaki Okumura
{"title":"Chebyshev’s bias for Fermat curves of prime degree","authors":"Yoshiaki Okumura","doi":"10.1007/s11139-024-00913-7","DOIUrl":"https://doi.org/10.1007/s11139-024-00913-7","url":null,"abstract":"<p>In this article, we prove that an asymptotic formula for the prime number race with respect to Fermat curves of prime degree is equivalent to part of the Deep Riemann Hypothesis (DRH), which is a conjecture on the convergence of partial Euler products of <i>L</i>-functions on the critical line. We also show that such an equivalence holds for some quotients of Fermat curves. As an application, we compute the order of zero at <span>(s=1)</span> for the second moment <i>L</i>-functions of those curves under DRH.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883629","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 distribution of $$alpha p$$ modulo one in the intersection of two Piatetski–Shapiro sets 论 $$alpha p$$ modulo one 在两个 Piatetski-Shapiro 集的交集中的分布
The Ramanujan Journal Pub Date : 2024-08-02 DOI: 10.1007/s11139-024-00914-6
Xiaotian Li, Jinjiang Li, Min Zhang
{"title":"On the distribution of $$alpha p$$ modulo one in the intersection of two Piatetski–Shapiro sets","authors":"Xiaotian Li, Jinjiang Li, Min Zhang","doi":"10.1007/s11139-024-00914-6","DOIUrl":"https://doi.org/10.1007/s11139-024-00914-6","url":null,"abstract":"<p>Let <span>(lfloor trfloor )</span> denote the integer part of <span>(tin mathbb {R})</span> and <span>(Vert xVert )</span> the distance from <i>x</i> to the nearest integer. Suppose that <span>(1/2&lt;gamma _2&lt;gamma _1&lt;1)</span> are two fixed constants. In this paper, it is proved that, whenever <span>(alpha )</span> is an irrational number and <span>(beta )</span> is any real number, there exist infinitely many prime numbers <i>p</i> in the intersection of two Piatetski–Shapiro sets, i.e., <span>(p=lfloor n_1^{1/gamma _1}rfloor =lfloor n_2^{1/gamma _2}rfloor )</span>, such that </p><span>$$begin{aligned} Vert alpha p+beta Vert &lt;p^{-frac{12(gamma _1+gamma _2)-23}{38}+varepsilon }, end{aligned}$$</span><p>provided that <span>(23/12&lt;gamma _1+gamma _2&lt;2)</span>. This result constitutes an generalization upon the previous result of Dimitrov (Indian J Pure Appl Math 54(3):858–867, 2023).</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883685","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
Some properties of totients 图腾的一些特性
The Ramanujan Journal Pub Date : 2024-08-02 DOI: 10.1007/s11139-024-00921-7
Pentti Haukkanen
{"title":"Some properties of totients","authors":"Pentti Haukkanen","doi":"10.1007/s11139-024-00921-7","DOIUrl":"https://doi.org/10.1007/s11139-024-00921-7","url":null,"abstract":"<p>A arithmetical function <i>f</i> is said to be a totient if there exist completely multiplicative functions <span>(f_t)</span> and <span>(f_v)</span> such that<span>( f=f_t*f_v^{-1}, )</span> where <span>(*)</span> is the Dirichlet convolution. Euler’s <span>(phi )</span>-function is an important example of a totient. In this paper we find the structure of the usual product of two totients, the usual integer power of totients, the usual product of a totient and a specially multiplicative function and the usual product of a totient and a completely multiplicative function. These results are derived with the aid of generating series. We also provide some distributive-like characterizations of totients involving the usual product and the Dirichlet convolution of arithmetical functions. They give as corollaries characterizations of completely multiplicative functions.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"144 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883686","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
Consecutive pure cubic fields with large class number 大类数连续纯立方场
The Ramanujan Journal Pub Date : 2024-08-01 DOI: 10.1007/s11139-024-00912-8
Dongho Byeon, Donggeon Yhee
{"title":"Consecutive pure cubic fields with large class number","authors":"Dongho Byeon, Donggeon Yhee","doi":"10.1007/s11139-024-00912-8","DOIUrl":"https://doi.org/10.1007/s11139-024-00912-8","url":null,"abstract":"<p>In this paper, we prove that for a given positive integer <i>k</i>, there are at least <span>(x^{1/3-o(1)})</span> integers <span>(d le x)</span> such that the consecutive pure cubic fields <span>({mathbb {Q}}(root 3 of {d+1}))</span>, <span>(cdots )</span>, <span>({mathbb {Q}}(root 3 of {d+k}))</span> have arbitrarily large class numbers.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"217 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883524","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
Ghost series and a motivated proof of the Bressoud–Göllnitz–Gordon identities 幽灵级数和布列索-哥尼兹-哥顿等式的动机证明
The Ramanujan Journal Pub Date : 2024-07-26 DOI: 10.1007/s11139-024-00908-4
John Layne, Samuel Marshall, Christopher Sadowski, Emily Shambaugh
{"title":"Ghost series and a motivated proof of the Bressoud–Göllnitz–Gordon identities","authors":"John Layne, Samuel Marshall, Christopher Sadowski, Emily Shambaugh","doi":"10.1007/s11139-024-00908-4","DOIUrl":"https://doi.org/10.1007/s11139-024-00908-4","url":null,"abstract":"<p>We present what we call a “motivated proof” of the Bressoud-Göllnitz-Gordon identities. Similar “motivated proofs” have been given by Andrews and Baxter for the Rogers–Ramanujan identities and by Lepowsky and Zhu for Gordon’s identities. Additionally, “motivated proofs” have also been given for the Andrews-Bressoud identities by Kanade, Lepowsky, Russell, and Sills and for the Göllnitz–Gordon–Andrews identities by Coulson, Kanade, Lepowsky, McRae, Qi, Russell, and the third author. Our proof borrows both the use of “ghost series” from the “motivated proof” of the Andrews–Bressoud identities and uses recursions similar to those found in the “motivated proof” of the Göllnitz–Gordon–Andrews identities. We anticipate that this “motivated proof” of the Bressoud–Göllnitz–Gordon identities will illuminate certain twisted vertex-algebraic constructions.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771495","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
Determining universality of m-gonal forms with first five coefficients 确定具有前五个系数的 monal 形式的普遍性
The Ramanujan Journal Pub Date : 2024-07-22 DOI: 10.1007/s11139-024-00849-y
Dayoon Park
{"title":"Determining universality of m-gonal forms with first five coefficients","authors":"Dayoon Park","doi":"10.1007/s11139-024-00849-y","DOIUrl":"https://doi.org/10.1007/s11139-024-00849-y","url":null,"abstract":"<p>We classify the <span>((a_1,a_2,a_3,a_4,a_5))</span> for which the universality of an <i>m</i>-gonal form <span>(F_m({textbf{x}}))</span> whose first five coefficients are <span>((a_1,a_2,a_3,a_4,a_5))</span> is characterized as the representabilitiy of positive integers up to <span>(m-4)</span> and discuss some applications.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"111 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771497","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 general divisor problem of higher moments of coefficients attached to the Dedekind zeta function 戴德金泽塔函数所附系数高矩数的一般除数问题
The Ramanujan Journal Pub Date : 2024-07-19 DOI: 10.1007/s11139-024-00907-5
Guodong Hua
{"title":"The general divisor problem of higher moments of coefficients attached to the Dedekind zeta function","authors":"Guodong Hua","doi":"10.1007/s11139-024-00907-5","DOIUrl":"https://doi.org/10.1007/s11139-024-00907-5","url":null,"abstract":"<p>Let <span>(K_{3})</span> be a non-normal cubic extension over <span>(mathbb {Q})</span>. And let <span>(tau _{k}^{K_{3}}(n))</span> denote the <i>k</i>-dimensional divisor function in the number field <span>(K_{3}/mathbb {Q})</span>. In this paper, we investigate the higher moments of the coefficients attached to the Dedekind zeta function over sum of two squares of the form </p><span>$$begin{aligned} sum _{n_{1}^{2}+n_{2}^{2}le x}(tau _{k}^{K_{3}}(n_{1}^{2}+n_{2}^{2}))^{l}, end{aligned}$$</span><p>where <span>(n_{1}, n_{2}in mathbb {Z})</span>, and <span>(kge 2, lge 2)</span> are any fixed integers.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741163","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
Purely periodic continued fractions and graph-directed iterated function systems 纯周期性续分和图形定向迭代函数系统
The Ramanujan Journal Pub Date : 2024-07-19 DOI: 10.1007/s11139-024-00904-8
Giovanni Panti
{"title":"Purely periodic continued fractions and graph-directed iterated function systems","authors":"Giovanni Panti","doi":"10.1007/s11139-024-00904-8","DOIUrl":"https://doi.org/10.1007/s11139-024-00904-8","url":null,"abstract":"<p>We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in <span>({{,textrm{P},}}^1mathbb {R})</span>, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, <span>(ldots )</span>) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"08 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741237","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
Exponential radii of starlikeness and convexity of some special functions 一些特殊函数的星形和凸形的指数半径
The Ramanujan Journal Pub Date : 2024-07-18 DOI: 10.1007/s11139-024-00902-w
Adiba Naz, Sumit Nagpal, V. Ravichandran
{"title":"Exponential radii of starlikeness and convexity of some special functions","authors":"Adiba Naz, Sumit Nagpal, V. Ravichandran","doi":"10.1007/s11139-024-00902-w","DOIUrl":"https://doi.org/10.1007/s11139-024-00902-w","url":null,"abstract":"<p>Using the Hadamard factorization, the exponential radii of starlikeness and convexity for various special functions like Wright function, Lommel function, Struve function, Ramanujan type entire function, cross product and product of Bessel function have been investigated. For certain ranges of the parameters appearing in these special functions, the precise values of the exponential radii of starlikeness and convexity are calculated as the solutions of transcendental equations. The interlacing property of the zeros of special functions and their derivatives is the fundamental technique utilized to demonstrate these results.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"889 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741239","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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