Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.013
Nilmoni Karak, Kamalakshya Mahatab
The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved Ω-bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.
{"title":"The Piltz divisor problem in number fields using the resonance method","authors":"Nilmoni Karak, Kamalakshya Mahatab","doi":"10.1016/j.jnt.2025.10.013","DOIUrl":"10.1016/j.jnt.2025.10.013","url":null,"abstract":"<div><div>The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved Ω-bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 726-740"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618277","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 : 2025-11-19DOI: 10.1016/j.jnt.2025.10.015
Li Lu, Lingyu Guo, Victor Zhenyu Guo
We prove a new bound to the exponential sum of the form by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form . In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in a Beatty sequence and so on.
我们用I型和的一种新方法证明了形式为∑h ~ h δh∑m ~ m∑n ~ Nmn ~ xambne（αmn+h(mn+u)γ）的指数和的一个新界。这个和可以应用于许多与皮亚茨基-夏皮罗素数有关的问题，它是形式为⌊nc⌋的素数。本文改进了Balog-Friedlander条件的可容许范围，从而改进了带Piatetski-Shapiro素数的三元哥德巴赫问题。我们还研究了等差数列中的Piatetski-Shapiro素数的分布，Beatty数列中的Piatetski-Shapiro素数等。
{"title":"Improvements on exponential sums related to Piatetski-Shapiro primes","authors":"Li Lu, Lingyu Guo, Victor Zhenyu Guo","doi":"10.1016/j.jnt.2025.10.015","DOIUrl":"10.1016/j.jnt.2025.10.015","url":null,"abstract":"<div><div>We prove a new bound to the exponential sum of the form<span><span><span><math><munder><mo>∑</mo><mrow><mi>h</mi><mo>∼</mo><mi>H</mi></mrow></munder><msub><mrow><mi>δ</mi></mrow><mrow><mi>h</mi></mrow></msub><munder><mrow><munder><mo>∑</mo><mrow><mi>m</mi><mo>∼</mo><mi>M</mi></mrow></munder><munder><mo>∑</mo><mrow><mi>n</mi><mo>∼</mo><mi>N</mi></mrow></munder></mrow><mrow><mi>m</mi><mi>n</mi><mo>∼</mo><mi>x</mi></mrow></munder><mspace></mspace><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>e</mi><mo>(</mo><mi>α</mi><mi>m</mi><mi>n</mi><mo>+</mo><mi>h</mi><msup><mrow><mo>(</mo><mi>m</mi><mi>n</mi><mo>+</mo><mi>u</mi><mo>)</mo></mrow><mrow><mi>γ</mi></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span> by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form <span><math><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>⌋</mo></math></span>. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in a Beatty sequence and so on.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 700-725"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618248","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 : 2025-11-19DOI: 10.1016/j.jnt.2025.10.011
Muhammad Afifurrahman
We obtain an asymptotic formula for the number of integer matrices that have determinant Δ and whose absolute values of the entries are at most H. The result holds uniformly for a large range of Δ with respect to H.
{"title":"A uniform formula on the number of integer matrices with given determinant and height","authors":"Muhammad Afifurrahman","doi":"10.1016/j.jnt.2025.10.011","DOIUrl":"10.1016/j.jnt.2025.10.011","url":null,"abstract":"<div><div>We obtain an asymptotic formula for the number of integer <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices that have determinant Δ and whose absolute values of the entries are at most <em>H</em>. The result holds uniformly for a large range of Δ with respect to <em>H</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 741-770"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617773","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 : 2025-11-19DOI: 10.1016/j.jnt.2025.10.016
Peter Vang Uttenthal
In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on that remain inequivalent modulo 3 for and 20887, and that these ℓ are the three smallest such primes.
{"title":"Parameters of solvable automorphic forms","authors":"Peter Vang Uttenthal","doi":"10.1016/j.jnt.2025.10.016","DOIUrl":"10.1016/j.jnt.2025.10.016","url":null,"abstract":"<div><div>In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>)</mo></math></span> that remain inequivalent modulo 3 for <span><math><mi>ℓ</mi><mo>=</mo><mn>7687</mn><mo>,</mo><mn>16363</mn></math></span> and 20887, and that these <em>ℓ</em> are the three smallest such primes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 771-794"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618275","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 : 2025-11-10DOI: 10.1016/j.jnt.2025.10.001
Weitong Wang
The proof of Lemma 7.10 is flawed. In this corrigendum, we prove the statement under additional conditions which do not affect the statistical results for -fields and -fields in the introduction part.
{"title":"Corrigendum to “Invariant part of class groups and distribution of relative class group” [J. Number Theory 279 (2026) 691–748]","authors":"Weitong Wang","doi":"10.1016/j.jnt.2025.10.001","DOIUrl":"10.1016/j.jnt.2025.10.001","url":null,"abstract":"<div><div>The proof of Lemma 7.10 is flawed. In this corrigendum, we prove the statement under additional conditions which do not affect the statistical results for <span><math><mo>(</mo><mn>6</mn><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span>-fields and <span><math><mo>(</mo><mn>3</mn><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-fields in the introduction part.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 588-595"},"PeriodicalIF":0.7,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520454","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 : 2025-11-07DOI: 10.1016/j.jnt.2025.10.008
Caleb M. Shor , Jae Hyung Sim
In 2008, Wang & Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers a and b is equidistributed modulo 2 precisely when a and b are both odd. Shor generalized this in 2022, showing that the set of gaps of such a numerical semigroup is equidistributed modulo m when a and b are coprime to m and at least one of them is 1 modulo m. In this paper, we further generalize these results by considering numerical semigroups generalized by geometric sequences of the form , aiming to determine when the corresponding set of gaps is equidistributed modulo m. With elementary methods, we are able to obtain a result for and all m. We then work with cyclotomic rings, using results about multiplicative independence of cyclotomic units to obtain results for all k and infinitely many m. Finally, we take an approach with cyclotomic units and Dirichlet L-functions to obtain results for all k and all m.
Wang &； Wang在2008年证明了当a和b都是奇数时，由两个素数正整数a和b生成的数值半群的间隙集精确地是等分布模2。Shor在2022年推广了这一结论，证明了当a和b对m互素且至少有一个是1模m时，这样的数值半群的间隙集是等分布模m。在本文中，我们进一步推广了这些结果，考虑了由ak，ak−1b，…，bk形式的几何序列广义的数值半群，目的是确定相应的间隙集何时是等分布模m。我们能够得到k=2和所有m的结果。然后我们使用环切环，使用关于环切单元乘法独立性的结果来获得所有k和无限多个m的结果。最后，我们采用环切单元和Dirichlet l -函数的方法来获得所有k和所有m的结果。
{"title":"Equidistribution conditions for gaps of geometric numerical semigroups","authors":"Caleb M. Shor , Jae Hyung Sim","doi":"10.1016/j.jnt.2025.10.008","DOIUrl":"10.1016/j.jnt.2025.10.008","url":null,"abstract":"<div><div>In 2008, Wang & Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers <em>a</em> and <em>b</em> is equidistributed modulo 2 precisely when <em>a</em> and <em>b</em> are both odd. Shor generalized this in 2022, showing that the set of gaps of such a numerical semigroup is equidistributed modulo <em>m</em> when <em>a</em> and <em>b</em> are coprime to <em>m</em> and at least one of them is 1 modulo <em>m</em>. In this paper, we further generalize these results by considering numerical semigroups generalized by geometric sequences of the form <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>b</mi><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, aiming to determine when the corresponding set of gaps is equidistributed modulo <em>m</em>. With elementary methods, we are able to obtain a result for <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> and all <em>m</em>. We then work with cyclotomic rings, using results about multiplicative independence of cyclotomic units to obtain results for all <em>k</em> and infinitely many <em>m</em>. Finally, we take an approach with cyclotomic units and Dirichlet L-functions to obtain results for all <em>k</em> and all <em>m</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 615-647"},"PeriodicalIF":0.7,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520453","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 : 2025-11-03DOI: 10.1016/j.jnt.2025.10.005
George Huang , Henry H. Kim
We compute the scattering constants of three families of noncongruence subgroups, using the Kronecker limit formula.
利用Kronecker极限公式计算了三族非同余子群的散射常数。
{"title":"Scattering constants for some families of noncongruence subgroups","authors":"George Huang , Henry H. Kim","doi":"10.1016/j.jnt.2025.10.005","DOIUrl":"10.1016/j.jnt.2025.10.005","url":null,"abstract":"<div><div>We compute the scattering constants of three families of noncongruence subgroups, using the Kronecker limit formula.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 533-561"},"PeriodicalIF":0.7,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467141","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 : 2025-11-03DOI: 10.1016/j.jnt.2025.10.003
Martí Oller
For families of curves arising from a Dynkin diagram of type ADE, we show that the density of such curves having squarefree discriminant is equal to the product of local densities. We do so using the framework of Thorne and Laga's PhD theses and geometry-of-numbers techniques developed by Bhargava, here expanded over number fields.
{"title":"Geometry-of-numbers over number fields and the density of ADE families of curves having squarefree discriminant","authors":"Martí Oller","doi":"10.1016/j.jnt.2025.10.003","DOIUrl":"10.1016/j.jnt.2025.10.003","url":null,"abstract":"<div><div>For families of curves arising from a Dynkin diagram of type ADE, we show that the density of such curves having squarefree discriminant is equal to the product of local densities. We do so using the framework of Thorne and Laga's PhD theses and geometry-of-numbers techniques developed by Bhargava, here expanded over number fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 492-532"},"PeriodicalIF":0.7,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467131","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 : 2025-11-03DOI: 10.1016/j.jnt.2025.10.004
Tim Gehrunger, Richard Pink
Consider a hyperelliptic curve of genus g over a field K of characteristic zero. After extending K we can view it as a marked curve with its Weierstrass points. We present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic 2 over a finite extension of K. In the cases we work out relatively simple conditions for the structure of this reduction.
{"title":"Reduction of hyperelliptic curves in residue characteristic 2","authors":"Tim Gehrunger, Richard Pink","doi":"10.1016/j.jnt.2025.10.004","DOIUrl":"10.1016/j.jnt.2025.10.004","url":null,"abstract":"<div><div>Consider a hyperelliptic curve of genus <em>g</em> over a field <em>K</em> of characteristic zero. After extending <em>K</em> we can view it as a marked curve with its <span><math><mn>2</mn><mi>g</mi><mo>+</mo><mn>2</mn></math></span> Weierstrass points. We present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic 2 over a finite extension of <em>K</em>. In the cases <span><math><mi>g</mi><mo>⩽</mo><mn>2</mn></math></span> we work out relatively simple conditions for the structure of this reduction.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 429-491"},"PeriodicalIF":0.7,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467142","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 : 2025-10-31DOI: 10.1016/j.jnt.2025.10.007
Pierre-Yves Bienvenu
In this note, we study the set of values of the quadruplet where and denote the lower and upper asymptotic density, respectively. Complementing existing results on the topic, we determine each of its six projections on coordinate planes, that is, the sets of possible values of the six subpairs of the quadruplet. Further, we show that this set has non empty interior, in particular has positive measure. To do so, we use among others probabilistic and diophantine methods. Some auxiliary results pertaining to these methods may be of general interest.
在这篇文章中，我们研究了四重态(d_(A), D (A),d_（2A), D (2A)）的值的集合D，其中A∧N和d_， D分别表示下和上渐近密度。补充已有的结果，我们确定了它在坐标平面上的六个投影，即四重体的六个子对的可能值的集合。进一步证明了该集合D具有非空的内部，特别是具有正测度。为此，我们使用概率和丢番图方法。与这些方法有关的一些辅助结果可能会引起一般的兴趣。
{"title":"Realisability of simultaneous density constraints for sets of integers","authors":"Pierre-Yves Bienvenu","doi":"10.1016/j.jnt.2025.10.007","DOIUrl":"10.1016/j.jnt.2025.10.007","url":null,"abstract":"<div><div>In this note, we study the set <span><math><mi>D</mi></math></span> of values of the quadruplet <span><math><mo>(</mo><munder><mrow><mi>d</mi></mrow><mo>_</mo></munder><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><munder><mrow><mi>d</mi></mrow><mo>_</mo></munder><mo>(</mo><mn>2</mn><mi>A</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mo>‾</mo></mover><mo>(</mo><mn>2</mn><mi>A</mi><mo>)</mo><mo>)</mo></math></span> where <span><math><mi>A</mi><mo>⊂</mo><mi>N</mi></math></span> and <span><math><munder><mrow><mi>d</mi></mrow><mo>_</mo></munder><mo>,</mo><mover><mrow><mi>d</mi></mrow><mo>‾</mo></mover></math></span> denote the lower and upper asymptotic density, respectively. Complementing existing results on the topic, we determine each of its six projections on coordinate planes, that is, the sets of possible values of the six subpairs of the quadruplet. Further, we show that this set <span><math><mi>D</mi></math></span> has non empty interior, in particular has positive measure. To do so, we use among others probabilistic and diophantine methods. Some auxiliary results pertaining to these methods may be of general interest.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 596-614"},"PeriodicalIF":0.7,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520452","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号