Journal of Number Theory最新文献

{"title":"The irrationality of an infinite series involving ω(n) under a prime tuples conjecture","authors":"Kyle Pratt","doi":"10.1016/j.jnt.2025.02.010","DOIUrl":"10.1016/j.jnt.2025.02.010","url":null,"abstract":"<div><div>Let <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of distinct prime factors of <em>n</em>. Assuming a suitably uniform version of the prime <em>k</em>-tuples conjecture, we show that the number<span><span><span><math><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mfrac><mrow><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac></mrow></math></span></span></span> is irrational. This settles (conditionally) a question of Erdős.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 57-71"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917290","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}

{"title":"Class numbers, congruent numbers and umbral moonshine","authors":"Miranda C.N. Cheng ,&nbsp;John F.R. Duncan ,&nbsp;Michael H. Mertens","doi":"10.1016/j.jnt.2025.02.007","DOIUrl":"10.1016/j.jnt.2025.02.007","url":null,"abstract":"<div><div>In earlier work we initiated a program to study relationships between finite groups and arithmetic geometric invariants of modular curves in a systematic way. In the present work we continue this program, with a focus on the two smallest sporadic simple Mathieu groups. To do this we first elucidate a connection between a special case of umbral moonshine and the imaginary quadratic class numbers. Then we use this connection to classify a distinguished set of modules for the smallest sporadic Mathieu group. Finally we establish a connection between our classification and the congruent number problem of antiquity.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 201-235"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069560","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}

{"title":"Restricted free products of Demushkin groups of rank ℵ0 as absolute Galois groups","authors":"Tamar Bar-On","doi":"10.1016/j.jnt.2025.03.012","DOIUrl":"10.1016/j.jnt.2025.03.012","url":null,"abstract":"<div><div>We prove that a restricted free profinite (pro-<em>p</em>) product over a countable set of pro-<em>p</em> Demushkin groups of rank <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, that can be realized as absolute Galois groups, is isomorphic to an absolute Galois group if and only if <span><math><msub><mrow><mi>log</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⁡</mo><mi>q</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 257-269"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935257","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}

{"title":"Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture","authors":"Victor Zhenyu Guo,&nbsp;Yuan Yi","doi":"10.1016/j.jnt.2025.02.008","DOIUrl":"10.1016/j.jnt.2025.02.008","url":null,"abstract":"<div><div>The Piatetski-Shapiro sequences are of the form <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></msup><mo>:</mo><mo>=</mo><msubsup><mrow><mo>(</mo><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>⌋</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> with <span><math><mi>c</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>∉</mo><mi>N</mi></math></span>. Let <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the sequence of primes in ascending order. In this paper, we study the distribution of pairs <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> of consecutive primes such that <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></msup></math></span> for <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and give a conjecture with the prime counting functions of the pairs <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>. We give a heuristic argument to support this prediction based on a model by Lemke Oliver and Soundararajan which relies on a strong form of the Hardy-Littlewood conjecture. Moreover, we prove a proposition related to the average of singular series with a weight of a complex exponential function.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 286-314"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943230","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}

{"title":"Rational deformations of the set of multiple zeta-star values","authors":"Jiangtao Li","doi":"10.1016/j.jnt.2025.03.004","DOIUrl":"10.1016/j.jnt.2025.03.004","url":null,"abstract":"<div><div>In this paper we study the derived sets for the rational deformations of multiple zeta-star values. By using the theory of bounded variation functions, we will give function decompositions which describe the metric structure of the derived sets. The connection between the rational deformation of multiple zeta-star values and the <em>n</em>-th Cantor set in fractal geometry is also discussed.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 23-56"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917289","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}

{"title":"The small value set polynomials over finite fields and monodromy groups","authors":"Xiantao Deng,&nbsp;Bin Xu,&nbsp;Qianxi Zhu","doi":"10.1016/j.jnt.2025.03.008","DOIUrl":"10.1016/j.jnt.2025.03.008","url":null,"abstract":"<div><div>In this article, we study polynomials over finite fields with small value sets through their monodromy groups. More specifically, we consider polynomials <span><math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span> such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo></math></span> is Galois, which we call absolutely minimal value set polynomials (AMVSPs) over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We examine their relationship with minimal value set polynomials (MVSPs) by directly utilizing tools and results from function field theory. In particular, we prove that if <span><math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span> satisfies <span><math><mn>1</mn><mo>&lt;</mo><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>f</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></math></span>, then <span><math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is an MVSP if and only if it is an AMVSP. As an application of our results, we provide a simple proof of the classification results of Mills (see <span><span>[16, Theorem 2]</span></span>) and Borges-Conceição (see <span><span>[4, Theorem 2.3]</span></span>) for MVSPs with degree <span><math><mo>≤</mo><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></math></span> using a completely different approach.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 139-161"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935259","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}

{"title":"Divisibility of the multiplicative order modulo monic irreducible polynomials over finite fields","authors":"Joaquim Cera Da Conceição","doi":"10.1016/j.jnt.2025.03.003","DOIUrl":"10.1016/j.jnt.2025.03.003","url":null,"abstract":"<div><div>We consider the set of monic irreducible polynomials <em>P</em> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> such that the multiplicative order modulo <em>P</em> of some <em>a</em> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is divisible by a fixed positive integer <em>d</em>. Call <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> this set. We show the existence or non-existence of the density of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> for three distinct notions of density. In particular, the sets <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> have a Dirichlet density. Under some assumptions, we prove simple formulas for the density values.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 105-123"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942531","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}

{"title":"Multiplicative generalised polynomial sequences","authors":"Jakub Konieczny","doi":"10.1016/j.jnt.2025.03.007","DOIUrl":"10.1016/j.jnt.2025.03.007","url":null,"abstract":"<div><div>We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the sequence is not zero almost everywhere.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 147-164"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942533","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}

{"title":"Maesaka–Seki–Watanabe's formula for multiple harmonic q-sums","authors":"Yuto Tsuruta","doi":"10.1016/j.jnt.2025.02.009","DOIUrl":"10.1016/j.jnt.2025.02.009","url":null,"abstract":"<div><div>Maesaka, Seki, and Watanabe recently discovered an equality called the MSW formula. This paper provides a <em>q</em>-analogue of MSW formula. It discusses a new proof of the duality relation for finite multiple harmonic <em>q</em>-series at primitive roots of unity via <em>q</em>-analogue of MSW formula. This paper also gives a <em>q</em>-analogue of Yamamoto's generalization of MSW formula for Schur type.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 270-285"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943228","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}

{"title":"Cyclotomic primes","authors":"Carl Pomerance","doi":"10.1016/j.jnt.2025.02.013","DOIUrl":"10.1016/j.jnt.2025.02.013","url":null,"abstract":"<div><div>Mersenne primes and Fermat primes may be thought of as primes of the form <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the <em>m</em>th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 198-208"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935256","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}