Journal of Number Theory最新文献

{"title":"Bushnell-Reiner zeta functions over two-dimensional semilocal rings","authors":"Sean B. Lynch","doi":"10.1016/j.jnt.2025.07.010","DOIUrl":"10.1016/j.jnt.2025.07.010","url":null,"abstract":"<div><div>Lustig gave an infinite product formula for the zeta function of a commutative two-dimensional regular local ring with finite residue field. We extend this to the noncommutative setting with a method based on filtration by an invertible ideal. One application gives an abstract two-dimensional analogue of Hey's formula. Another application provides effective formulae for zeta functions over Rump's two-dimensional regular semiperfect rings. In the appendices, we supplement these two-dimensional applications with requisite one-dimensional calculations.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 1-34"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144921775","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 square-root law does not hold in the presence of zero divisors","authors":"Nathaniel Kingsbury-Neuschotz","doi":"10.1016/j.jnt.2025.08.020","DOIUrl":"10.1016/j.jnt.2025.08.020","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;R&lt;/em&gt; be a finite ring (with identity, not necessarily commutative) and define the paraboloid &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of &lt;em&gt;P&lt;/em&gt; satisfies a square-root law of the form &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for all nontrivial additive characters &lt;em&gt;ψ&lt;/em&gt;, with &lt;em&gt;C&lt;/em&gt; some fixed constant (for instance, if &lt;em&gt;R&lt;/em&gt; is a finite field, this bound will be satisfied with &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;). Then all but finitely many of the rings are fields.&lt;/div&gt;&lt;div&gt;Most of our argument works in greater generality: let &lt;em&gt;f&lt;/em&gt; be a polynomial with integer coefficients in &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; variables, with a fixed order of variable multiplications (so that it defines a function &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; even when &lt;em&gt;R&lt;/em&gt; is noncommutative), and set &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, then all but finitely many of the rings are fields or matrix rings of small dimension. We also describe how our techniques can establish that certain varieties do not satisfy a square root law ","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 481-505"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220605","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":"Spherical varieties and non-ordinary families of cohomology classes","authors":"Rob Rockwood","doi":"10.1016/j.jnt.2025.08.012","DOIUrl":"10.1016/j.jnt.2025.08.012","url":null,"abstract":"<div><div>We show that <em>p</em>-adic families of cohomology classes associated to symmetric spaces vary <em>p</em>-adically over small discs in weight space, without any ordinarity assumption. This generalises previous work of Loeffler, Zerbes and the author. Furthermore, we show that these families exhibit full variation in the cyclotomic direction, generalising previous constructions of Euler systems and <em>p</em>-adic <em>L</em>-functions. As an application we show that the Lemma–Flach Euler system of Loeffler–Skinner–Zerbes interpolates in Coleman families.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 390-454"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220703","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":"2026-03-01","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":"Infinitude of the zeros of the Lerch zeta function on the half plane ℜ(s)>1","authors":"Biswajyoti Saha ,&nbsp;Dhananjaya Sahu","doi":"10.1016/j.jnt.2025.08.019","DOIUrl":"10.1016/j.jnt.2025.08.019","url":null,"abstract":"<div><div>For <span><math><mi>a</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, the zeros of the Hurwitz zeta function <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup></math></span> have interesting features. There are no zeros in the half plane <span><math><mo>ℜ</mo><mo>(</mo><mi>s</mi><mo>)</mo><mo>≥</mo><mn>1</mn><mo>+</mo><mi>a</mi></math></span>, whereas there are infinitely many zeros in the strip <span><math><mn>1</mn><mo>&lt;</mo><mo>ℜ</mo><mo>(</mo><mi>s</mi><mo>)</mo><mo>&lt;</mo><mn>1</mn><mo>+</mo><mi>a</mi></math></span>, provided <span><math><mi>a</mi><mo>≠</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>1</mn></math></span>. The existence of these infinitely many zeros was first proved by Davenport and Heilbronn for rational and transcendental values of <em>a</em> and then by Cassels for algebraic irrational values of <em>a</em>. In this article, we consider the analogous question for the zeros of the cognate Lerch zeta function <span><math><msub><mrow><mi>ζ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup></math></span>, where <em>z</em> is a complex number of unit modulus. When <em>z</em> is a root of unity, the question can be answered using a theorem of Zaghloul, which is an extension of a work of Chatterjee and Gun. In the general case, we need to further extend the method of Chatterjee and Gun.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 506-518"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220656","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":"Corresponding Abelian extensions of integrally equivalent number fields","authors":"Shaver Phagan","doi":"10.1016/j.jnt.2025.08.001","DOIUrl":"10.1016/j.jnt.2025.08.001","url":null,"abstract":"<div><div>Extensive work has been done to determine necessary and sufficient conditions for a bijective correspondence of abelian extensions of number fields to force an isomorphism of the base fields. However, explicit examples of correspondences over non-isomorphic fields are rare. Integrally equivalent number fields admit an induced correspondence of abelian extensions. Studying this correspondence using idelic class field theory and linear algebra, we show that the corresponding extensions share features similar to those of arithmetically equivalent fields, and yet they are not generally weakly Kronecker equivalent. We also extend a group cohomological result of Arapura-Katz-McReynolds-Solapurkar and present geometric and arithmetic applications.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 88-112"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060280","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":"2026-03-01","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":"2026-03-01","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":"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":"2026-03-01","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":"2026-03-01","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}