Pub Date : 2026-03-01Epub Date: 2025-10-01DOI: 10.1016/j.jnt.2025.09.003
Sara Chari , Sergio Ricardo Zapata Ceballos , Erik Holmes , Fatemeh Jalalvand , Rahinatou Yuh Njah Nchiwo , Kelly O'Connor , Fabian Ramirez , Sameera Vemulapalli
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.
In this work we focus on -quartic fields with signature ; such fields have a rank 2 unit group. Viewing the unit lattice as a point of , we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of . Moreover, we produce three explicit (algebraic) points of which are limit points of the set of (points associated to) unit lattices of -quartic fields with signature .
{"title":"Unit lattices of D4-quartic number fields with signature (2,1)","authors":"Sara Chari , Sergio Ricardo Zapata Ceballos , Erik Holmes , Fatemeh Jalalvand , Rahinatou Yuh Njah Nchiwo , Kelly O'Connor , Fabian Ramirez , 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":"2026-03-01","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}
Pub Date : 2026-03-01Epub Date: 2025-09-24DOI: 10.1016/j.jnt.2025.08.020
Nathaniel Kingsbury-Neuschotz
<div><div>Let <em>R</em> be a finite ring (with identity, not necessarily commutative) and define the paraboloid <span><math><mi>P</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>}</mo></math></span>. Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of <em>P</em> satisfies a square-root law of the form <span><math><mo>|</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>ψ</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>C</mi><mo>|</mo><mi>R</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mo>|</mo><mi>P</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> for all nontrivial additive characters <em>ψ</em>, with <em>C</em> some fixed constant (for instance, if <em>R</em> is a finite field, this bound will be satisfied with <span><math><mi>C</mi><mo>=</mo><mn>1</mn></math></span>). Then all but finitely many of the rings are fields.</div><div>Most of our argument works in greater generality: let <em>f</em> be a polynomial with integer coefficients in <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span> variables, with a fixed order of variable multiplications (so that it defines a function <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><mi>R</mi></math></span> even when <em>R</em> is noncommutative), and set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>}</mo></math></span>. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span>, 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
{"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":"<div><div>Let <em>R</em> be a finite ring (with identity, not necessarily commutative) and define the paraboloid <span><math><mi>P</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>}</mo></math></span>. Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of <em>P</em> satisfies a square-root law of the form <span><math><mo>|</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>ψ</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>C</mi><mo>|</mo><mi>R</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mo>|</mo><mi>P</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> for all nontrivial additive characters <em>ψ</em>, with <em>C</em> some fixed constant (for instance, if <em>R</em> is a finite field, this bound will be satisfied with <span><math><mi>C</mi><mo>=</mo><mn>1</mn></math></span>). Then all but finitely many of the rings are fields.</div><div>Most of our argument works in greater generality: let <em>f</em> be a polynomial with integer coefficients in <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span> variables, with a fixed order of variable multiplications (so that it defines a function <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><mi>R</mi></math></span> even when <em>R</em> is noncommutative), and set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>}</mo></math></span>. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span>, 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}
Pub Date : 2026-03-01Epub Date: 2025-09-24DOI: 10.1016/j.jnt.2025.08.012
Rob Rockwood
We show that p-adic families of cohomology classes associated to symmetric spaces vary p-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 p-adic L-functions. As an application we show that the Lemma–Flach Euler system of Loeffler–Skinner–Zerbes interpolates in Coleman families.
{"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}
Pub Date : 2026-03-01Epub Date: 2025-10-02DOI: 10.1016/j.jnt.2025.09.010
Helen W.J. Zhang, Ying Zhong
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.
{"title":"Asymptotics and limiting distributions of several overpartition statistics","authors":"Helen W.J. Zhang, 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}
Pub Date : 2026-03-01Epub Date: 2025-09-23DOI: 10.1016/j.jnt.2025.08.019
Biswajyoti Saha , Dhananjaya Sahu
For , the zeros of the Hurwitz zeta function have interesting features. There are no zeros in the half plane , whereas there are infinitely many zeros in the strip , provided . The existence of these infinitely many zeros was first proved by Davenport and Heilbronn for rational and transcendental values of a and then by Cassels for algebraic irrational values of a. In this article, we consider the analogous question for the zeros of the cognate Lerch zeta function , where z is a complex number of unit modulus. When z 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.
{"title":"Infinitude of the zeros of the Lerch zeta function on the half plane ℜ(s)>1","authors":"Biswajyoti Saha , 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><</mo><mo>ℜ</mo><mo>(</mo><mi>s</mi><mo>)</mo><mo><</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}
Pub Date : 2026-03-01Epub Date: 2025-09-04DOI: 10.1016/j.jnt.2025.08.001
Shaver Phagan
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.
{"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}
Pub Date : 2026-03-01Epub Date: 2025-09-04DOI: 10.1016/j.jnt.2025.08.004
Petr Kucheriaviy
<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo></mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1
如果自然数A中的任何元素都不能整除其他自然数，则称自然数A为本数。设Ω(n)为n具有多重性的质因数个数。设fz(A)=∑A∈AzΩ(A) A (log (A) z，其中z∈R>；0。Erdős在1935年证明了f1(A)=∑A∈A1alog (A)在所有原始集A上是一致有界的。我们证明了Erdős不等式的一个推广，对于z∈(0,2)时的fz(A)提供了一个类似的结果。进一步，我们研究了fz(A)在所有原始集合上的最优性。我们还讨论了limz→0 (A)，它是Dirichlet密度的推广。我们研究了fz（Pk）的渐近性，其中Pk={n：Ω(n)=k}。对于z=1，我们找到f1（Pk）的渐近展开中的下一项，改进了Gorodetsky， Lichtman和Wong的结果。我们还研究了∑a∈AzΩ(a)/a在[1，N]的所有原始子集上的最优性。
{"title":"Erdős inequality for primitive sets","authors":"Petr Kucheriaviy","doi":"10.1016/j.jnt.2025.08.004","DOIUrl":"10.1016/j.jnt.2025.08.004","url":null,"abstract":"<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo></mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 113-152"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096121","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}
Pub Date : 2026-03-01Epub Date: 2025-09-04DOI: 10.1016/j.jnt.2025.08.003
Paul M. Voutier
We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, , arising from the solutions of generalised negative Pell equations, , where −c and are any positive squares. We show that there are at most 2 distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most 5 distinct squares when for infinitely many values of b, including all , as well as once d exceeds an explicit lower bound, without any conditions on the size of such squares.
{"title":"Bounds on the number of squares in recurrence sequences: y0 = b2 (I)","authors":"Paul M. Voutier","doi":"10.1016/j.jnt.2025.08.003","DOIUrl":"10.1016/j.jnt.2025.08.003","url":null,"abstract":"<div><div>We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mo>∞</mo></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span>, arising from the solutions of generalised negative Pell equations, <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>d</mi><msup><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>c</mi></math></span>, where −<em>c</em> and <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are any positive squares. We show that there are at most 2 distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most 5 distinct squares when <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for infinitely many values of <em>b</em>, including all <span><math><mn>1</mn><mo>≤</mo><mi>b</mi><mo>≤</mo><mn>24</mn></math></span>, as well as once <em>d</em> exceeds an explicit lower bound, without any conditions on the size of such squares.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 246-270"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096118","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}
Pub Date : 2026-03-01Epub Date: 2025-10-02DOI: 10.1016/j.jnt.2025.09.011
Bartosz Sobolewski , Lukas Spiegelhofer
Let denote the number of ones in the binary expansion of the nonnegative integer n. How does behave under addition of a constant t? In order to study the differences for all , we consider the associated characteristic function . Our main theorem is a structural result on the decomposition of into a sum of components. We also study in detail the case that t contains at most two blocks of consecutive 1s. The results in this paper are motivated by Cusick's conjecture on the sum-of-digits function. This conjecture is concerned with the central tendency of the corresponding probability distributions, and is still unsolved.
{"title":"Decomposing the sum-of-digits correlation measure","authors":"Bartosz Sobolewski , 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}
Pub Date : 2026-03-01Epub Date: 2025-10-02DOI: 10.1016/j.jnt.2025.09.007
Mingyu Kim
A triangular form is an integer-valued quadratic polynomial of the form , where the coefficients are positive integers and . 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.
{"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}
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