Journal of Number Theory最新文献

{"title":"M-functions and screw functions: Applications to Goldbach's problem and zeros of the Riemann zeta-function","authors":"Kohji Matsumoto ,&nbsp;Masatoshi Suzuki","doi":"10.1016/j.jnt.2025.09.013","DOIUrl":"10.1016/j.jnt.2025.09.013","url":null,"abstract":"<div><div>We study the <em>M</em>-functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a sufficient condition for the Riemann hypothesis provided by some formulas of the <em>M</em>-functions, which was a necessary condition in previous work. The other new aspect is the relation between the secondary main terms and the screw functions, which provides another necessary and sufficient condition for the Riemann hypothesis. We study such <em>M</em>-functions and screw functions in generalized settings by axiomatizing them.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 918-946"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262238","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":"Hilbert modular Eisenstein congruences of local origin","authors":"Dan Fretwell ,&nbsp;Jenny Roberts","doi":"10.1016/j.jnt.2025.09.009","DOIUrl":"10.1016/j.jnt.2025.09.009","url":null,"abstract":"<div><div>Let <em>F</em> be an arbitrary totally real field. Under standard conditions we prove the existence of certain Eisenstein congruences between parallel weight <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> Hilbert eigenforms of level <span><math><mi>mp</mi></math></span> and Hilbert Eisenstein series of level <span><math><mi>m</mi></math></span>, for arbitrary ideal <span><math><mi>m</mi></math></span> and prime ideal <span><math><mi>p</mi><mo>∤</mo><mi>m</mi></math></span> of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span>. Such congruences have their moduli coming from special values of Hecke <em>L</em>-functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving general results about this.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 861-896"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267053","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":"Asymptotics and limiting distributions of several overpartition statistics","authors":"Helen W.J. Zhang,&nbsp;Ying Zhong","doi":"10.1016/j.jnt.2025.09.010","DOIUrl":"10.1016/j.jnt.2025.09.010","url":null,"abstract":"<div><div>This paper primarily is dedicated to studying the asymptotics and limiting distributions of several statistics in overpartitions. As a preliminary result, we use asymptotic methods to prove that the number of distinct parts and distinct integers in overpartitions is asymptotically normal, extending the results of Corteel and Hitczenko. Furthermore, we investigate the asymptotic and distributional properties of two types of crank statistics for overpartitions, originally introduced by Bringmann and Lovejoy. Utilizing the Hardy-Ramanujan circle method, we derive asymptotic formulas for the moments of these two cranks, as well as for the symmetrized moments proposed by Jennings-Shaffer. Building on these, we employ the probabilistic method of moments to prove that both two cranks asymptotically follow a logistic distribution when appropriately normalized. Consequently, our results recover the asymptotic formulas for the positive moments first obtained by Zapata Rolon using Wright's circle method.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 737-760"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267058","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":"Decomposing the sum-of-digits correlation measure","authors":"Bartosz Sobolewski ,&nbsp;Lukas Spiegelhofer","doi":"10.1016/j.jnt.2025.09.011","DOIUrl":"10.1016/j.jnt.2025.09.011","url":null,"abstract":"<div><div>Let <span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of ones in the binary expansion of the nonnegative integer <em>n</em>. How does <span><math><mi>s</mi></math></span> behave under addition of a constant <em>t</em>? In order to study the differences<span><span><span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>+</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>,</mo></math></span></span></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>, we consider the associated characteristic function <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>. Our main theorem is a structural result on the decomposition of <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> into a sum of <em>components</em>. We also study in detail the case that <em>t</em> contains at most two blocks of consecutive <span>1</span>s. The results in this paper are motivated by <em>Cusick's conjecture</em> on the sum-of-digits function. This conjecture is concerned with the <em>central tendency</em> of the corresponding probability distributions, and is still unsolved.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 702-736"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267057","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":"Regular triangular forms of rank exceeding 3","authors":"Mingyu Kim","doi":"10.1016/j.jnt.2025.09.007","DOIUrl":"10.1016/j.jnt.2025.09.007","url":null,"abstract":"<div><div>A triangular form is an integer-valued quadratic polynomial of the form <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where the coefficients <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> are positive integers and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. A triangular form is called regular if it represents all positive integers which are locally represented. In this article, we determine all regular triangular forms of more than three variables.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 825-860"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267056","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":"On members of Lucas sequences with bounded prime gaps","authors":"Attila Bérczes ,&nbsp;Lajos Hajdu ,&nbsp;Florian Luca ,&nbsp;István Pink","doi":"10.1016/j.jnt.2025.09.005","DOIUrl":"10.1016/j.jnt.2025.09.005","url":null,"abstract":"<div><div>In this paper, we look at terms of Lucas sequences whose prime factors have indices with bounded gaps in the sequence of all prime numbers. Some of our results depend on certain widely believed conjectures. In our proofs we combine various tools, including Baker's method, the subspace theorem, and results of Stewart, and Murty and Wong.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 897-917"},"PeriodicalIF":0.7,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262237","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":"Unit lattices of D4-quartic number fields with signature (2,1)","authors":"Sara Chari ,&nbsp;Sergio Ricardo Zapata Ceballos ,&nbsp;Erik Holmes ,&nbsp;Fatemeh Jalalvand ,&nbsp;Rahinatou Yuh Njah Nchiwo ,&nbsp;Kelly O'Connor ,&nbsp;Fabian Ramirez ,&nbsp;Sameera Vemulapalli","doi":"10.1016/j.jnt.2025.09.003","DOIUrl":"10.1016/j.jnt.2025.09.003","url":null,"abstract":"<div><div>There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results.</div><div>In this work we focus on <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-quartic fields with signature <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>; such fields have a rank 2 unit group. Viewing the unit lattice as a point of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>h</mi></math></span>, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>h</mi></math></span>. Moreover, we produce three explicit (algebraic) points of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>h</mi></math></span> which are limit points of the set of (points associated to) unit lattices of <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-quartic fields with signature <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 761-784"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267055","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":"An optimal lower bound for the size of the restricted sumsets containing powers","authors":"Wang-Xing Yu ,&nbsp;Jun-Jia Zhao","doi":"10.1016/j.jnt.2025.09.001","DOIUrl":"10.1016/j.jnt.2025.09.001","url":null,"abstract":"<div><div>Let <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> be a fixed real number and <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> be an integer. In 2023, Yu, Chen and Chen proved that for any sufficiently large positive integer <em>n</em>, if <span><math><mi>A</mi><mo>⊆</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span> with <span><math><mi>gcd</mi><mo>⁡</mo><mi>A</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>&gt;</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>m</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span>, then there is a power of <em>r</em> that can be represented as the sum of distinct elements of <em>A</em>, where <span><math><mi>m</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> is a computable positive integer only related to <em>r</em>. In this paper, we improve this result for <span><math><mi>r</mi><mo>≥</mo><mn>3</mn></math></span>. We prove that the condition <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>&gt;</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>m</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> can be replaced by <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>&gt;</mo><mi>n</mi><mo>/</mo><mi>m</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span>, where <span><math><mi>f</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> is a computable positive integer only related to <em>r</em>. We will also show that this lower bound is optimal, namely, for infinitely many positive integers <em>n</em>, there exists <span><math><mi>B</mi><mo>⊆</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span> with <span><math><mi>gcd</mi><mo>⁡</mo><mi>B</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mo>|</mo><mi>B</mi><mo>|</mo><mo>=</mo><mi>n</mi><mo>/</mo><mi>m</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> such that no power of <em>r</em> can be represented as the sum of distinct elements of <em>B</em>. This also generalizes a result in which <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span> obtained by Yang and Zhao.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 785-807"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267054","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":"Initial layer of the anti-cyclotomic Zp-extension of Q(−m) and capitulation phenomenon","authors":"Georges Gras","doi":"10.1016/j.jnt.2025.09.004","DOIUrl":"10.1016/j.jnt.2025.09.004","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mi>m</mi></mrow></msqrt><mo>)</mo></math></span> be an imaginary quadratic field. We consider the properties of capitulation of the <em>p</em>-class group of <em>k</em> in the anti-cyclotomic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup></math></span> of <em>k</em>; for this, using a new approach based on the <span><math><msub><mrow><mi>Log</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-function (<span><span>Theorem 2.3</span></span>, <span><span>Theorem 3.4</span></span>), we determine the first layer <span><math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math></span> of <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup></math></span> over <em>k</em>, and we show that some partial capitulation may exist in <span><math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math></span>, even when <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup><mo>/</mo><mi>k</mi></math></span> is totally ramified. We have conjectured that this phenomenon of capitulation is specific of the <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extensions of <em>k</em>, distinct from the cyclotomic one. For <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>, we characterize a sub-family of fields <em>k</em> (Normal Split cases) for which <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>ac</mi></mrow></msup></math></span> is not linearly disjoint from the Hilbert class field (<span><span>Theorem 5.1</span></span>). No assumptions are made on the splitting of 3 in <em>k</em> and in <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mn>3</mn><mi>m</mi></mrow></msqrt><mo>)</mo></math></span>, nor on the structures of their 3-class groups. Four <span>pari/gp</span> programs (<span><span>7.1</span></span>, <span><span>7.2</span></span>, <span><span>7.3</span></span>, <span><span>7.4</span></span> depending on the classification of <span><span>Definition 2.10</span></span>) are given, computing a defining cubic polynomial of <span><math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math></span>, and the main invariants attached to the fields <em>k</em>, <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, <span><math><msubsup><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>ac</mi></mrow></msubsup></math></span>; some relations with Iwasawa's invariants are discussed (<span><span>Theorem 9.6</span></span>).</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 634-701"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267235","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":"Comparing regular and backward continued fractions: Lochs-type theorems and approximation properties","authors":"Zhigang Tian ,&nbsp;Lulu Fang","doi":"10.1016/j.jnt.2025.09.006","DOIUrl":"10.1016/j.jnt.2025.09.006","url":null,"abstract":"<div><div>In this paper, we study two problems concerning the relationship between regular continued fractions (RCFs) and backward continued fractions (BCFs). The first problem addresses Lochs-type theorems for RCFs and BCFs, where we compare the number of partial quotients in one expansion as a function of the number of partial quotients in the other expansion. The second problem investigates the approximation properties of RCFs and BCFs, with particular attention to the set of irrational numbers that are infinitely often better approximated by BCFs than by RCFs. We show that this set has Lebesgue measure zero and further analyze it from the perspectives of Baire category and fractal dimension.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 947-972"},"PeriodicalIF":0.7,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267042","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}