Pub Date : 2024-08-22DOI: 10.1016/j.jnt.2024.07.010
Yin Choi Cheng
The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in that does not contain with . Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form to arbitrary linear forms ; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when for some , and also when , the “structured” domain. We also contrast the “enigmatic” domain when with the “structured” domain, and give upper bounds on the growth rates in both cases.
{"title":"Greedy Sidon sets for linear forms","authors":"Yin Choi Cheng","doi":"10.1016/j.jnt.2024.07.010","DOIUrl":"10.1016/j.jnt.2024.07.010","url":null,"abstract":"<div><p>The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in <span><math><mi>N</mi></math></span> that does not contain <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> with <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> to arbitrary linear forms <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for some <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, and also when <span><math><mi>h</mi><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mn>4</mn></math></span>, the “structured” domain. We also contrast the “enigmatic” domain when <span><math><mi>h</mi><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>3</mn></math></span> with the “structured” domain, and give upper bounds on the growth rates in both cases.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 225-248"},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001768/pdfft?md5=530dddb3b9f53a0f7a336819d6924b12&pid=1-s2.0-S0022314X24001768-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097369","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}
Pub Date : 2024-08-21DOI: 10.1016/j.jnt.2024.07.012
Kwok Chi Chim
We prove explicit lower bounds for linear forms in two p-adic logarithms. More specifically, we establish explicit lower bounds for the p-adic distance between two integral powers of algebraic numbers, that is, (and corresponding explicit upper bounds for ), where are numbers that are algebraic over and are positive rational integers.
This work is a p-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on and ) is , instead of as in the work of Bugeaud and Laurent in 1996.
{"title":"Lower bounds for linear forms in two p-adic logarithms","authors":"Kwok Chi Chim","doi":"10.1016/j.jnt.2024.07.012","DOIUrl":"10.1016/j.jnt.2024.07.012","url":null,"abstract":"<div><p>We prove explicit lower bounds for linear forms in two <em>p</em>-adic logarithms. More specifically, we establish explicit lower bounds for the <em>p</em>-adic distance between two integral powers of algebraic numbers, that is, <span><math><mo>|</mo><mi>Λ</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mo>|</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><msub><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> (and corresponding explicit upper bounds for <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>Λ</mi><mo>)</mo></math></span>), where <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are numbers that are algebraic over <span><math><mi>Q</mi></math></span> and <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are positive rational integers.</p><p>This work is a <em>p</em>-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>Λ</mi><mo>)</mo></math></span> has an explicit constant of reasonable size and the dependence of the bound on <em>B</em> (a quantity depending on <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>) is <span><math><mi>log</mi><mo></mo><mi>B</mi></math></span>, instead of <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>B</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> as in the work of Bugeaud and Laurent in 1996.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 295-349"},"PeriodicalIF":0.6,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001793/pdfft?md5=ccf251a8e8e82101b493968e4e90bf5e&pid=1-s2.0-S0022314X24001793-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162052","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.001
Ron Evans , Franz Lemmermeyer , Zhi-Hong Sun , Mark Van Veen
For squarefree , let M denote the ring class field for the order in . Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of such that E and F have the same discriminant. Define the real cube roots and , where is the fundamental unit in . We prove that E can be taken as if and only if . As byproducts of the proof, we give explicit congruences for a and b which hold if and only if , and we also show that the norm of the relative discriminant of lies in or according as or . We then prove that v is always in the ring class field for the order in F. Some of the results above are extended for subsets of properly containing the fundamental units .
对于无平方 d>1,让 M 表示 F=Q(-3d) 中阶 Z[-3d] 的环类域。哈塞证明,当且仅当存在一个 Q 的立方扩展 E,使得 E 和 F 具有相同的判别式时,3 平分 F 的类数。定义实立方根 v=(a+bd)1/3 和 v′=(a-bd)1/3,其中 a+bd 是 Q(d) 的基本单位。我们证明,当且仅当 v∈M 时,E 可以看作 Q(v+v′)。作为证明的副产品,我们给出了 a 和 b 的明确同余式,当且仅当 v∈M 时,这两个同余式成立,我们还证明了 F(v)/F 的相对判别式的规范位于{1,36}或{38,318},视 v∈M 或 v∉M 而定。然后,我们证明 v 总是在 F 的阶 Z[-27d] 的环类域中。上面的一些结果可以扩展到适当包含基本单元 a+bd 的 Q(d) 子集。
{"title":"Ring class fields and a result of Hasse","authors":"Ron Evans , Franz Lemmermeyer , Zhi-Hong Sun , Mark Van Veen","doi":"10.1016/j.jnt.2024.07.001","DOIUrl":"10.1016/j.jnt.2024.07.001","url":null,"abstract":"<div><p>For squarefree <span><math><mi>d</mi><mo>></mo><mn>1</mn></math></span>, let <em>M</em> denote the ring class field for the order <span><math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mn>3</mn><mi>d</mi></mrow></msqrt><mo>]</mo></math></span> in <span><math><mi>F</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>. Hasse proved that 3 divides the class number of <em>F</em> if and only if there exists a cubic extension <em>E</em> of <span><math><mi>Q</mi></math></span> such that <em>E</em> and <em>F</em> have the same discriminant. Define the real cube roots <span><math><mi>v</mi><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span>, where <span><math><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt></math></span> is the fundamental unit in <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>. We prove that <em>E</em> can be taken as <span><math><mi>Q</mi><mo>(</mo><mi>v</mi><mo>+</mo><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span> if and only if <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span>. As byproducts of the proof, we give explicit congruences for <em>a</em> and <em>b</em> which hold if and only if <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span>, and we also show that the norm of the relative discriminant of <span><math><mi>F</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>/</mo><mi>F</mi></math></span> lies in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>}</mo></math></span> or <span><math><mo>{</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>8</mn></mrow></msup><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>18</mn></mrow></msup><mo>}</mo></math></span> according as <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span> or <span><math><mi>v</mi><mo>∉</mo><mi>M</mi></math></span>. We then prove that <em>v</em> is always in the ring class field for the order <span><math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mn>27</mn><mi>d</mi></mrow></msqrt><mo>]</mo></math></span> in <em>F</em>. Some of the results above are extended for subsets of <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span> properly containing the fundamental units <span><math><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 33-61"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001677/pdfft?md5=4a76de3ef7096a558707691b3467bc3b&pid=1-s2.0-S0022314X24001677-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012867","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.008
Robert D. Hough , Eun Hye Lee
Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip.
在此之前,作者证明了枚举二元三次方形式类数的新谷 zeta 函数的次凸性。在此,我们再次证明马斯形式扭曲版本的次凸性。这里演示的方法可应用于佐藤 F. 提出的一些扭曲zeta函数的次凸性。这一论证表明,佐藤所使用的对称空间条件对于估计临界带中的zeta函数并非必要。
{"title":"Subconvexity of twisted Shintani zeta functions","authors":"Robert D. Hough , Eun Hye Lee","doi":"10.1016/j.jnt.2024.07.008","DOIUrl":"10.1016/j.jnt.2024.07.008","url":null,"abstract":"<div><p>Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 62-97"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001781/pdfft?md5=416d6328f418d63f0962779de94e173a&pid=1-s2.0-S0022314X24001781-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142021482","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.013
John Cullinan , Shanna Dobson , Linda Frey , Asimina S. Hamakiotes , Roberto Hernandez , Nathan Kaplan , Jorge Mello , Gabrielle Scullard
Let E and be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if but . Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.
继[6]之后,如果 E(Fp)≃E′(Fp),但 E(Fp2)≄E′(Fp2),我们就称好还原的素数 p 为反常素数。我们的主要结果是为任何这样一对椭圆曲线的反常素数比例提供了一个明确的公式。我们同时考虑了 CM 情况和非 CM 情况。
{"title":"The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II","authors":"John Cullinan , Shanna Dobson , Linda Frey , Asimina S. Hamakiotes , Roberto Hernandez , Nathan Kaplan , Jorge Mello , Gabrielle Scullard","doi":"10.1016/j.jnt.2024.07.013","DOIUrl":"10.1016/j.jnt.2024.07.013","url":null,"abstract":"<div><p>Let <em>E</em> and <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> be 2-isogenous elliptic curves over <strong>Q</strong>. Following <span><span>[6]</span></span>, we call a prime of good reduction <em>p anomalous</em> if <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>≃</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> but <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo><mo>≄</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math></span>. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 131-165"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001720/pdfft?md5=f8f53d9d54ebb568a03018d889d8244b&pid=1-s2.0-S0022314X24001720-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083985","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.009
S.V. Nagaraj
This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number of Carmichael numbers up to X with three prime factors is for all . He showed that his conjecture is true for and . In this article, it is shown that the conjecture is true for and . In both cases, analytical methods establish the conjecture for large X and tables of Carmichael numbers are used for small X.
本文描述了在实现 S.W. Graham 猜想方面取得的进展。他猜想,对于所有 X≥1 的卡迈克尔数 C3(X),直到 X 的三个质因数为 ≤X。他证明了他的猜想对于 X≤1016 和 X>10126 为真。本文则证明猜想在 X≤1024 和 X>2⁎1040 时为真。在这两种情况下,对于大的 X,都用分析方法建立了猜想,而对于小的 X,则使用了卡迈克尔数表。
{"title":"Progress towards a conjecture of S.W. Graham","authors":"S.V. Nagaraj","doi":"10.1016/j.jnt.2024.07.009","DOIUrl":"10.1016/j.jnt.2024.07.009","url":null,"abstract":"<div><p>This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of Carmichael numbers up to <em>X</em> with three prime factors is <span><math><mo>≤</mo><msqrt><mrow><mi>X</mi></mrow></msqrt></math></span> for all <span><math><mi>X</mi><mo>≥</mo><mn>1</mn></math></span>. He showed that his conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>></mo><msup><mrow><mn>10</mn></mrow><mrow><mn>126</mn></mrow></msup></math></span>. In this article, it is shown that the conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>></mo><mn>2</mn><mo>⁎</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>40</mn></mrow></msup></math></span>. In both cases, analytical methods establish the conjecture for large <em>X</em> and tables of Carmichael numbers are used for small <em>X</em>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 281-294"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001756/pdfft?md5=a96d01f7aa1e622c98ed012747b85804&pid=1-s2.0-S0022314X24001756-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097231","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.004
Yuda Chen, Xiangjun Dai, Huixi Li
Let be the set of positive odd numbers that can not be written in the form . Recently, by analyzing possible prime divisors of b, Chen proved and if an arithmetic progression is in , with if and only if , where is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove and provide all possible values of a if is in . Moreover, we explicitly construct nontrivial arithmetic progressions in with , 9, 10, or 11, and provide potential nontrivial arithmetic progressions in such that for any fixed . Furthermore, we improve the upper bound estimate of numbers of the form by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.
设 U 是不能写成 p+2k 形式的正奇数的集合。最近,Chen 通过分析 b 的可能素除数,证明了如果算术级数 a(modb) 在 U 中,则 b≥11184810 且 ω(b)≥7,当且仅当 b=11184810 时,ω(b)=7,其中 ω(n) 是 n 的不同素除数的个数。本文采用计算方法证明 b≥11184810,并提供了 a(mod11184810) 在 U 中时 a 的所有可能值。此外,我们明确地构造了 U 中 ω(b)=8, 9, 10 或 11 的非微不足道的算术级数 a(modb),并提供了 U 中潜在的非微不足道的算术级数 a(modb),使得任何固定的 s≥12 时,ω(b)=s。此外,我们通过增强算法和使用 GPU 计算,将哈布西格和罗布洛 2006 年对 p+2k 形式数的估计上限提高到了 0.490341088858244。
{"title":"Some computational results on a conjecture of de Polignac about numbers of the form p + 2k","authors":"Yuda Chen, Xiangjun Dai, Huixi Li","doi":"10.1016/j.jnt.2024.07.004","DOIUrl":"10.1016/j.jnt.2024.07.004","url":null,"abstract":"<div><p>Let <span><math><mi>U</mi></math></span> be the set of positive odd numbers that can not be written in the form <span><math><mi>p</mi><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. Recently, by analyzing possible prime divisors of <em>b</em>, Chen proved <span><math><mi>b</mi><mo>≥</mo><mn>11184810</mn></math></span> and <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>≥</mo><mn>7</mn></math></span> if an arithmetic progression <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>b</mi><mo>)</mo></math></span> is in <span><math><mi>U</mi></math></span>, with <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>=</mo><mn>7</mn></math></span> if and only if <span><math><mi>b</mi><mo>=</mo><mn>11184810</mn></math></span>, where <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the number of distinct prime divisors of <em>n</em>. In this paper, we take a computational approach to prove <span><math><mi>b</mi><mo>≥</mo><mn>11184810</mn></math></span> and provide all possible values of <em>a</em> if <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>11184810</mn><mo>)</mo></math></span> is in <span><math><mi>U</mi></math></span>. Moreover, we explicitly construct nontrivial arithmetic progressions <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>b</mi><mo>)</mo></math></span> in <span><math><mi>U</mi></math></span> with <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>=</mo><mn>8</mn></math></span>, 9, 10, or 11, and provide potential nontrivial arithmetic progressions <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>b</mi><mo>)</mo></math></span> in <span><math><mi>U</mi></math></span> such that <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>=</mo><mi>s</mi></math></span> for any fixed <span><math><mi>s</mi><mo>≥</mo><mn>12</mn></math></span>. Furthermore, we improve the upper bound estimate of numbers of the form <span><math><mi>p</mi><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span> by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 249-268"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001690/pdfft?md5=34d1d69eca5fb0f5a9878d3392dfc7c6&pid=1-s2.0-S0022314X24001690-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097312","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.002
Huixi Li , Biao Wang , Chunlin Wang , Shaoyun Yi
A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most 1016. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity s in any global function field of genus g over for . In particular, there is no covering system of with distinct moduli for .
{"title":"On Erdős covering systems in global function fields","authors":"Huixi Li , Biao Wang , Chunlin Wang , Shaoyun Yi","doi":"10.1016/j.jnt.2024.07.002","DOIUrl":"10.1016/j.jnt.2024.07.002","url":null,"abstract":"<div><p>A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most 10<sup>16</sup>. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to <span><math><mn>616</mn><mo>,</mo><mn>000</mn></math></span> by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity <em>s</em> in any global function field of genus <em>g</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for <span><math><mi>q</mi><mo>≥</mo><mo>(</mo><mn>1.14</mn><mo>+</mo><mn>0.16</mn><mi>g</mi><mo>)</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>6.5</mn><mo>+</mo><mn>0.97</mn><mi>g</mi></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. In particular, there is no covering system of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> with distinct moduli for <span><math><mi>q</mi><mo>≥</mo><mn>759</mn></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 269-280"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001707/pdfft?md5=f62f3fbb627421563f5c92d4564888ee&pid=1-s2.0-S0022314X24001707-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097370","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.003
Hershy Kisilevsky, Jungbae Nam
We consider heuristic predictions for small non-zero algebraic central values of twists of the L-function of an elliptic curve by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to extended to chosen families of cyclic extensions of fixed degree.
我们考虑了关于椭圆曲线 E/Q 的 L 函数捻的非零代数中心值的启发式预测。我们提供了这些预测的计算证据,以及这些预测对与 E/Q 相关的布劳尔-西格尔定理的实例的影响,该定理扩展到固定度数的循环扩展的选定族。
{"title":"Non-zero central values of Dirichlet twists of elliptic L-functions","authors":"Hershy Kisilevsky, Jungbae Nam","doi":"10.1016/j.jnt.2024.07.003","DOIUrl":"10.1016/j.jnt.2024.07.003","url":null,"abstract":"<div><p>We consider heuristic predictions for small non-zero algebraic central values of twists of the <em>L</em>-function of an elliptic curve <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> extended to chosen families of cyclic extensions of fixed degree.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 166-194"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001719/pdfft?md5=a0edf993c65449e6ef9e685a75b7c9ac&pid=1-s2.0-S0022314X24001719-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097371","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}
Pub Date : 2024-08-20DOI: 10.1016/j.jnt.2024.07.007
Nobuki Takeda
We give sufficient conditions for the vector-valued Kurokawa-Mizumoto congruence related to the Klingen-Eisenstein series to hold. We also give a reinterpretation for differential operators on automorphic forms by the representation theory.
{"title":"Kurokawa-Mizumoto congruence and differential operators on automorphic forms","authors":"Nobuki Takeda","doi":"10.1016/j.jnt.2024.07.007","DOIUrl":"10.1016/j.jnt.2024.07.007","url":null,"abstract":"<div><p>We give sufficient conditions for the vector-valued Kurokawa-Mizumoto congruence related to the Klingen-Eisenstein series to hold. We also give a reinterpretation for differential operators on automorphic forms by the representation theory.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"266 ","pages":"Pages 98-130"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001689/pdfft?md5=e293d3f84e4efad68424f0d31bf2f6e3&pid=1-s2.0-S0022314X24001689-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076717","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}