Journal of Number Theory最新文献_第3页

Ramanujan's continued fractions of order 10 as modular functions 拉马努金的10阶连分数作为模函数

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-06-04 DOI: 10.1016/j.jnt.2025.04.001

Victor Manuel Aricheta, Russelle Guadalupe

{"title":"Ramanujan's continued fractions of order 10 as modular functions","authors":"Victor Manuel Aricheta, Russelle Guadalupe","doi":"10.1016/j.jnt.2025.04.001","DOIUrl":"10.1016/j.jnt.2025.04.001","url":null,"abstract":"<div><div>We explore the modularity of the continued fractions <math><mi>I</mi><mo>(</mo><mi>τ</mi><mo>)</mo><mo>,</mo><mi>J</mi><mo>(</mo><mi>τ</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>τ</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>τ</mi><mo>)</mo></math>, and <math><mi>U</mi><mo>(</mo><mi>τ</mi><mo>)</mo><mo>=</mo><mi>I</mi><mo>(</mo><mi>τ</mi><mo>)</mo><mo>/</mo><mi>J</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> of order 10, which are special cases of certain identities of Ramanujan. The continued fractions <math><mi>I</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> and <math><mi>J</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> were recently introduced by Rajkhowa and Saikia. We show that these continued fractions can be expressed in terms of an η-quotient <math><mi>g</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> that generates the field of all modular functions on the congruence subgroup <math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>10</mn><mo>)</mo></math>. Consequently, we deduce that the modular equations for <math><mi>g</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> and <math><mi>U</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math> exist at any level and derive these equations of prime levels <math><mi>p</mi><mo>≤</mo><mn>11</mn></math>. We also show that the continued fractions of order 10 can be explicitly evaluated using a singular value of <math><mi>g</mi><mo>(</mo><mi>τ</mi><mo>)</mo></math>, which under certain conditions generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 214-244"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Weil-Barsotti formula for T-modules t模的Weil-Barsotti公式

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-06-04 DOI: 10.1016/j.jnt.2025.04.013

Dawid E. Kędzierski, Piotr Krasoń

{"title":"Weil-Barsotti formula for T-modules","authors":"Dawid E. Kędzierski, Piotr Krasoń","doi":"10.1016/j.jnt.2025.04.013","DOIUrl":"10.1016/j.jnt.2025.04.013","url":null,"abstract":"<div><div>In the work of M. A. Papanikolas and N. Ramachandran (2003) [26] the Weil-Barsotti formula for the function field case concerning <math><msubsup><mrow><mi>Ext</mi></mrow><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><mi>E</mi><mo>,</mo><mi>C</mi><mo>)</mo></math> where E is a Drinfeld module and C is the Carlitz module was proved. We generalize this formula to the case where E is a strictly pure t-module Φ with the zero nilpotent matrix <math><msub><mrow><mi>N</mi></mrow><mrow><mi>Φ</mi></mrow></msub></math>. For such a t-module Φ we explicitly compute its dual t-module <math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>∨</mo></mrow></msup></math> as well as its double dual <math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>∨</mo><mo>∨</mo></mrow></msup></math>. This computation is done in a subtle way by combination of the t-reduction algorithm developed by F. Głoch, D.E. Kędzierski, P. Krasoń, [arXiv:2408.08207<svg><path></path></svg>] [13] and the methods of the work of D.E. Kędzierski and P. Krasoń (2024) [20]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix <math><msub><mrow><mi>N</mi></mrow><mrow><mi>Φ</mi></mrow></msub></math> is non-zero.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 1-25"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144243011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Class numbers of binary quadratic polynomials 二元二次多项式的类数

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-06-04 DOI: 10.1016/j.jnt.2025.04.012

Zichen Yang

引用次数: 0

A note on zero-density approaches for the difference between consecutive primes 关于连续素数之差的零密度方法的注释

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-06-04 DOI: 10.1016/j.jnt.2025.04.007

Valeriia Starichkova

{"title":"A note on zero-density approaches for the difference between consecutive primes","authors":"Valeriia Starichkova","doi":"10.1016/j.jnt.2025.04.007","DOIUrl":"10.1016/j.jnt.2025.04.007","url":null,"abstract":"<div><div>In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem [12] which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is due to Heath-Brown and Iwaniec [9], which derives the weighted zero-density estimates used for obtaining the lower bound for the number of primes in short intervals. The generalised versions of these results make the connections between the zero-free regions, zero-density estimates, and the primes in short intervals more transparent. As an example, the generalisation of Ingham's theorem implies that, under the Density Hypothesis, the prime number theorem holds in <math><mo>[</mo><mi>x</mi><mo>−</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mi>exp</mi><mo>⁡</mo><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>⁡</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>x</mi><mo>]</mo></math>, which refines upon the classic interval <math><mo>[</mo><mi>x</mi><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>,</mo><mi>x</mi><mo>]</mo></math>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 245-266"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Quadratic forms in prime variables with small off–diagonal ranks 具有小非对角线列的素数变量的二次型

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-06-04 DOI: 10.1016/j.jnt.2025.04.006

Jakub Dobrowolski

引用次数: 0

Proofs of four conjectures of Ballantine, Feigon and Merca on linear inequalities of partitions with odd parts balantine、Feigon和Merca关于奇部分区线性不等式的四个猜想的证明

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-05-14 DOI: 10.1016/j.jnt.2025.03.014

Olivia X.M. Yao

引用次数: 0

Hecke relations for eta multipliers and congruences for higher-order smallest parts functions 乘子的Hecke关系和高阶最小部函数的同余

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-05-13 DOI: 10.1016/j.jnt.2025.03.006

Clayton Williams

引用次数: 0

On the number of zeros of L−functions attached to cusp forms of half-integral weight 半积分权值的尖形上的L−函数的零点数

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-05-09 DOI: 10.1016/j.jnt.2025.02.014

Pedro Ribeiro

引用次数: 0

Upper bounds for the lowest first zero in families of cuspidal newforms 尖形新形科中最低首零的上界

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-05-09 DOI: 10.1016/j.jnt.2025.02.012

Palak Arora , Glenn Bruda , Bruce Fang , Raul Marquez , Steven J. Miller , Beni Prapashtica , Vismay Sharan , Daeyoung Son , Xueyiming Tang , Saad Waheed

引用次数: 0

Distribution of cycles in supersingular ℓ-isogeny graphs 超奇异等构图中环的分布

IF 0.6 3区数学

Journal of Number Theory Pub Date : 2025-05-07 DOI: 10.1016/j.jnt.2025.03.013

Eli Orvis

引用次数: 0