Journal of Number Theory最新文献

{"title":"Note on a theorem of Birch–Erdős and m-ary partitions","authors":"Yuchen Ding ,&nbsp;Honghu Liu ,&nbsp;Zi Wang","doi":"10.1016/j.jnt.2025.07.009","DOIUrl":"10.1016/j.jnt.2025.07.009","url":null,"abstract":"<div><div>Let <span><math><mi>p</mi><mo>,</mo><mi>q</mi><mo>&gt;</mo><mn>1</mn></math></span> be two relatively prime integers and <span><math><mi>N</mi></math></span> the set of nonnegative integers. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of different expressions of <em>n</em> written as a sum of distinct terms taken from <span><math><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>:</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span>. Erdős conjectured and then Birch proved that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math></span> provided that <em>n</em> is sufficiently large. In this note, for all sufficiently large number <em>n</em> we prove<span><span><span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mi>p</mi><mi>log</mi><mo>⁡</mo><mi>q</mi></mrow></mfrac><mo>(</mo><mn>1</mn><mo>+</mo><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></msup><mo>.</mo></math></span></span></span> We also show that <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Additionally, we will point out the relations between <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <em>m</em>-ary partitions.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 910-928"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908202","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":"Exponential shrinking problem in multiplicative Diophantine approximation","authors":"Qi Jia,&nbsp;Junjie Shi","doi":"10.1016/j.jnt.2025.07.016","DOIUrl":"10.1016/j.jnt.2025.07.016","url":null,"abstract":"<div><div>Besides limsup set, the liminf set also appears widely in Diophantine approximation. It gives precise information about when a point can be well approximated compared with limsup set. Moreover, one usually uses liminf set to determine the dimension of limsup set from below. In this paper, we consider the liminf setting within the context of multiplicative Diophantine approximation. More precisely, let <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be a sequence of positive integers with exponential growth speed. For any <span><math><mi>τ</mi><mo>&gt;</mo><mn>0</mn></math></span>, define<span><span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>τ</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>:</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></munderover><mo>‖</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>‖</mo><mo>≤</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>τ</mi></mrow></msubsup><mspace></mspace><mspace></mspace><mrow><mi>for all</mi></mrow><mspace></mspace><mspace></mspace><mi>n</mi><mspace></mspace><mrow><mi>ultimately</mi></mrow><mo>}</mo></mrow><mo>.</mo></math></span></span></span> Hausdorff dimension of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>τ</mi><mo>)</mo></math></span> is presented in this note.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 969-986"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144925789","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":"Proof of the complete presence of a modulo 4 bias for the semiprimes","authors":"Miroslav Marinov ,&nbsp;Nikola Gyulev","doi":"10.1016/j.jnt.2025.07.004","DOIUrl":"10.1016/j.jnt.2025.07.004","url":null,"abstract":"<div><div>In 2016, Dummit, Granville, and Kisilevsky showed that the proportion of semiprimes (products of two primes) not exceeding a given <em>x</em>, whose factors are congruent to 3 modulo 4, is more than a quarter when <em>x</em> is sufficiently large. They have also conjectured that this holds from the very beginning, that is, for all <span><math><mi>x</mi><mo>≥</mo><mn>9</mn></math></span>. Here we give a proof of this conjecture. For <span><math><mi>x</mi><mo>≥</mo><mn>1.1</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>13</mn></mrow></msup></math></span> we take an explicit approach based on their work. We rely on classical estimates for prime counting functions, as well as on very recent explicit improvements by Bennett, Martin, O'Bryant, and Rechnitzer, which have wide applications in essentially any setting involving estimations of sums over primes in arithmetic progressions. All <span><math><mi>x</mi><mo>&lt;</mo><mn>1.1</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>13</mn></mrow></msup></math></span> are covered by a computed assisted verification.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 777-791"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904486","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":"Ramification filtration via deformations, II","authors":"Victor Abrashkin","doi":"10.1016/j.jnt.2025.07.005","DOIUrl":"10.1016/j.jnt.2025.07.005","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a field of formal Laurent series with coefficients in a finite field of characteristic &lt;em&gt;p&lt;/em&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the maximal quotient of the Galois group of &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of period &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and nilpotent class &lt;&lt;em&gt;p&lt;/em&gt; and let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the filtration by ramification subgroups in upper numbering. We use the identification &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of nilpotent Artin-Schreier theory: here &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the group obtained from a suitable profinite Lie &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; via the Campbell-Hausdorff composition law. We develop new techniques to obtain a “geometrical” construction of the ideals &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;. Given &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;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we construct a decreasing central filtration &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;⩽&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, on &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, an epimorphism of Lie &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-algebras &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;†&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⟶&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and a unipotent action Ω of &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; on &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;†&lt;/m","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 651-690"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904482","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":"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":"2025-08-21","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":"Variance of point-counts for families of cubic curves over Fp and Jacobsthal sums","authors":"Bogdan Nica","doi":"10.1016/j.jnt.2025.07.014","DOIUrl":"10.1016/j.jnt.2025.07.014","url":null,"abstract":"<div><div>We compute the variance of the number of points along one-parameter families of cubic curves. We highlight explicit evaluations of variances that make use of Jacobsthal sums.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 603-625"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902299","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":"Mass distribution for holomorphic cusp forms on the vertical geodesic","authors":"Qingfeng Sun ,&nbsp;Qizhi Zhang","doi":"10.1016/j.jnt.2025.07.011","DOIUrl":"10.1016/j.jnt.2025.07.011","url":null,"abstract":"<div><div>We compute the quantum variance of holomorphic cusp forms on the vertical geodesic for smooth compactly supported test functions. As an application we show that almost all holomorphic Hecke cusp forms, whose weights are in a short interval, satisfy QUE conjecture on the vertical geodesic.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 827-857"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907929","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":"Transcendental nature of p-adic Euler-Lehmer constants","authors":"Tapas Chatterjee ,&nbsp;Sonam Garg","doi":"10.1016/j.jnt.2025.07.018","DOIUrl":"10.1016/j.jnt.2025.07.018","url":null,"abstract":"<div><div>Murty and Saradha (2008) initiated a significant exploration into the transcendental nature of certain <em>p</em>-adic constants, focusing on the <em>p</em>-adic analogues of the Euler's constant <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and the Euler-Lehmer constant <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>/</mo><mi>p</mi><mo>)</mo></math></span>, where <em>p</em> is a rational prime with <span><math><mn>1</mn><mo>≤</mo><mi>r</mi><mo>&lt;</mo><mi>p</mi></math></span>. Their work laid the foundation for understanding these constants in the context of <em>p</em>-adic analysis.</div><div>This investigation was subsequently expanded by Chatterjee and Gun (2014), who extended the study to encompass the case of sets of prime numbers. In this article, we build upon their findings by generalizing the results further to include prime powers and products of prime powers. Our primary focus is to delve deeper into the transcendental properties of the <em>p</em>-adic analogues of the Euler-Lehmer constants in this broader framework.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 761-776"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904485","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":"Injectivity of the genus 1 Kudla–Millson lift on locally symmetric spaces","authors":"Ingmar Metzler ,&nbsp;Riccardo Zuffetti","doi":"10.1016/j.jnt.2025.07.017","DOIUrl":"10.1016/j.jnt.2025.07.017","url":null,"abstract":"<div><div>Let <em>L</em> be an even indefinite lattice. We show that if <em>L</em> splits off a hyperbolic plane and a scaled hyperbolic plane, then the Kudla–Millson lift of genus 1 associated to <em>L</em> is injective. Our result includes as special cases all previously known injectivity results on the whole space of elliptic cusp forms available in the literature. In particular, we also consider the Funke–Millson twist of the lift. Further, we provide geometric applications on locally symmetric spaces of orthogonal type.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 792-826"},"PeriodicalIF":0.7,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907140","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 m-th element of a Sidon set","authors":"R. Balasubramanian ,&nbsp;Sayan Dutta","doi":"10.1016/j.jnt.2025.07.007","DOIUrl":"10.1016/j.jnt.2025.07.007","url":null,"abstract":"<div><div>We prove that if <span><math><mi>A</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow></msub><mo>}</mo><mo>⊂</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> is a Sidon set so that <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, then<span><span><span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mi>m</mi><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>7</mn><mo>/</mo><mn>8</mn></mrow></msup><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>)</mo></mrow></math></span></span></span> where <span><math><mi>L</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mn>0</mn><mo>,</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>}</mo></math></span>. As an application of this, we give easy proofs of some previously derived results. We proceed on to proving that for a dense Sidon set <em>S</em> and for any <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span>, we have<span><span><span><math><munder><mo>∑</mo><mrow><mi>a</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>a</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>11</mn><mo>/</mo><mn>8</mn></mrow></msup><mo>)</mo></mrow></math></span></span></span> for all <span><math><mi>n</mi><mo>≤</mo><mi>N</mi></math></span> but at most <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>ε</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></mrow></math></span> exceptions.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 594-602"},"PeriodicalIF":0.7,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902298","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}