Journal of Number Theory最新文献

{"title":"Conditional estimates for L-functions in the Selberg class","authors":"Neea Palojärvi ,&nbsp;Aleksander Simonič","doi":"10.1016/j.jnt.2025.12.014","DOIUrl":"10.1016/j.jnt.2025.12.014","url":null,"abstract":"<div><div>Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of <span><math><mrow><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>/</mo><mi>L</mi><mo>)</mo></mrow><mo>(</mo><mi>s</mi><mo>)</mo></math></span> and <span><math><mi>log</mi><mo>⁡</mo><mi>L</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> for <span><math><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>δ</mi><mo>≤</mo><mi>σ</mi><mo>&lt;</mo><mn>1</mn></math></span>, fixed <span><math><mi>δ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></math></span> and for functions in the Selberg class. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> on prime numbers. Moreover, by assuming a polynomial Euler product representation for <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span>, we establish uniform bounds for <span><math><mo>|</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>−</mo><mi>σ</mi><mo>|</mo><mo>≤</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></math></span>, <span><math><mo>|</mo><mn>1</mn><mo>−</mo><mi>σ</mi><mo>|</mo><mo>≤</mo><mn>1</mn><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></math></span> and <span><math><mi>σ</mi><mo>=</mo><mn>1</mn></math></span>, and completely explicit estimates by assuming also the strong <em>λ</em>-conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 135-193"},"PeriodicalIF":0.7,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192989","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":"Counting rational points on Hirzebruch–Kleinschmidt varieties over global function fields","authors":"Sebastián Herrero ,&nbsp;Tobías Martínez ,&nbsp;Pedro Montero","doi":"10.1016/j.jnt.2026.01.008","DOIUrl":"10.1016/j.jnt.2026.01.008","url":null,"abstract":"<div><div>Inspired by Bourqui's work on anticanonical height zeta functions on Hirzebruch surfaces, we study height zeta functions of complete smooth split toric varieties with Picard rank 2 over global function fields, with respect to height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where precise analytic properties of the corresponding height zeta functions can be given. As application, we obtain asymptotic formulas for the number of rational points of large height on each subvariety, with explicit leading constants and controlled error terms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 1-53"},"PeriodicalIF":0.7,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147485","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":"Direct and inverse problems for restricted signed sumsets","authors":"Raj Kumar Mistri,&nbsp;Nitesh Prajapati","doi":"10.1016/j.jnt.2025.12.013","DOIUrl":"10.1016/j.jnt.2025.12.013","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&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;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be a nonempty finite subset of an additive abelian group &lt;em&gt;G&lt;/em&gt;. For a positive integer &lt;em&gt;h&lt;/em&gt;, the &lt;em&gt;h-fold signed sumset of A&lt;/em&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, is defined as&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&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;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;for&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;and&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and the &lt;em&gt;restricted h-fold signed sumset of A&lt;/em&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∧&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, is defined as&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∧&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;for&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;and&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; A direct problem for the sumset &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∧&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is to find the optimal size of &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∧&lt;/mo&gt;&lt;/","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 74-134"},"PeriodicalIF":0.7,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192988","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":"Simply connectedness and hyperbolicity","authors":"Carlo Gasbarri ,&nbsp;Erwan Rousseau ,&nbsp;Amos Turchet ,&nbsp;Julie Tzu-Yueh Wang","doi":"10.1016/j.jnt.2026.01.005","DOIUrl":"10.1016/j.jnt.2026.01.005","url":null,"abstract":"<div><div>We generalize to arbitrary dimension our previous construction of simply connected weakly-special but not special varieties. We show that they satisfy the function field and complex analytic part of Campana's conjecture. Moreover, we give examples, in any dimension, of smooth simply connected nonisotrivial projective varieties of general type that satisfy the function field Lang and Vojta conjectures with an explicit exceptional set.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 194-208"},"PeriodicalIF":0.7,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192987","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":"Modular supercuspidal lifts of weight 2","authors":"Iván Blanco-Chacón ,&nbsp;Luis Dieulefait","doi":"10.1016/j.jnt.2025.12.015","DOIUrl":"10.1016/j.jnt.2025.12.015","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi><mo>/</mo><mi>Q</mi></math></span> be a totally real number field and <span><math><mi>N</mi></math></span> an ideal of its ring of integers of norm <em>N</em>. Let <span><math><mi>p</mi><mo>&gt;</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>}</mo></math></span> be a prime totally split in <em>F</em> such that <span><math><mi>p</mi><mo>∤</mo><mi>N</mi></math></span>. For every even <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, define the <span><math><mo>[</mo><mi>F</mi><mo>:</mo><mi>Q</mi><mo>]</mo></math></span>-dimensional parallel weight <span><math><mtext>k</mtext><mo>=</mo><mo>(</mo><mi>k</mi><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>k</mi><mo>)</mo></math></span>. Let <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext>k</mtext></mrow></msub><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>)</mo></math></span> be any non CM Hilbert cuspidal Hecke eigenform. Assume that the residual representation <span><math><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>f</mi><mo>,</mo><mi>P</mi></mrow></msub></math></span> has large image for some prime <span><math><mi>P</mi></math></span> over <em>p</em> in the field of definition of <em>f</em>. Under these conditions, we prove that there exists a lift of <span><math><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>f</mi><mo>,</mo><mi>P</mi></mrow></msub></math></span> associated to a Hilbert modular cuspform <span><math><mi>g</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext>2</mtext></mrow></msub><mo>(</mo><mi>N</mi><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>ϵ</mi><mo>)</mo></math></span> which is supercuspidal at each prime of <em>F</em> over <em>p</em>. We also give a proof of the corresponding statement for classical Hecke cuspforms. Such statement was already proved by Khare <span><span>[23]</span></span> with classical techniques. Finally, using our main result we give a corrigenda for <span><span>[12]</span></span>, correctly inserting the micro good dihedral prime in the level.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 54-73"},"PeriodicalIF":0.7,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the existence of zero-sum subsequences with length not divided by a given number","authors":"Weidong Gao ,&nbsp;Xiao Jiang ,&nbsp;Yuanlin Li ,&nbsp;Huijuan Qi","doi":"10.1016/j.jnt.2025.12.002","DOIUrl":"10.1016/j.jnt.2025.12.002","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group and <em>k</em> be an integer not dividing the exponent of <em>G</em>. We denote by <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> the smallest positive integer <em>l</em> such that every sequence over <em>G</em> of length no less than <em>l</em> has a zero-sum subsequence of length not divisible by <em>k</em>. In this paper, we focus on determining <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, a cyclic group of order <em>n</em>. Specifically, we prove that<span><span><span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>⌊</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span></span></span> for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>}</mo><mo>∪</mo><mo>(</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 15-37"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080811","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":"Multiple standard twists of L-functions","authors":"J. Kaczorowski ,&nbsp;A. Perelli","doi":"10.1016/j.jnt.2026.01.003","DOIUrl":"10.1016/j.jnt.2026.01.003","url":null,"abstract":"<div><div>The standard twist of <em>L</em>-functions plays a fundamental role in the Selberg class theory. It is defined as an absolutely convergent Dirichlet series and admits meromorphic continuation beyond the half-plane of absolute convergence. Nowadays, the analytic properties of the standard twist <span><math><mi>F</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> of an <em>L</em>-function <em>F</em> are well-understood. For example, it has poles when the positive number <em>α</em> belongs to the so-called spectrum of <em>F</em>, and is entire otherwise. In this paper, for a given set <span><math><mi>F</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>}</mo></math></span> of <em>L</em>-functions and <span><math><mi>s</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, we consider the multiple standard twist <span><math><mi>F</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span>. This is defined initially on a certain half-space of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, and we describe its meromorphic continuation to the whole space. Results in the multidimensional case are, in many ways, analogous to those in the one-dimensional case. In particular, the spectrum of a multiple standard twist is relevant to the description of the set of poles of <span><math><mi>F</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span>. There are also significant differences; for instance, in the structure of the singularities.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 188-213"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190613","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 distribution of square-free integers in arithmetic progressions with prime power moduli","authors":"Mingxuan Zhong ,&nbsp;Tianping Zhang","doi":"10.1016/j.jnt.2026.01.006","DOIUrl":"10.1016/j.jnt.2026.01.006","url":null,"abstract":"<div><h3>Text</h3><div>We obtain an asymptotic formula for the distribution of square-free integers <span><math><mi>k</mi><mo>≤</mo><mi>X</mi></math></span> in an arithmetic progression <span><math><mi>k</mi><mo>≡</mo><mi>a</mi><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mi>q</mi><mo>)</mo></math></span> uniformly for <span><math><mi>q</mi><mo>≤</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>16</mn><mo>−</mo><mi>Δ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>Δ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is a decreasing function of <em>n</em>. Specially, when <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, we can get <span><math><mn>1</mn><mo>/</mo><mn>16</mn><mo>−</mo><mi>Δ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mn>1</mn><mo>/</mo><mn>1044</mn></math></span>. Previously Hooley (1975) showed that the asymptotic formula holds for a positive proportion of moduli <span><math><mi>q</mi><mo>≤</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn><mo>−</mo><mi>ε</mi></mrow></msup></math></span>, while for the first time Mangerel (2021) broke the well-known 3/4-barrier for <span><math><mi>q</mi><mo>≤</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>/</mo><mn>1044</mn><mo>−</mo><mi>ε</mi></mrow></msup></math></span> in the case of square-free, smooth moduli. Our results break the 3/4-barrier again in another case of prime power moduli and improve upon the range of <em>q</em> from the work of Mangerel. As a direct application, we derive a new record for the upper bound of the least square-free integer in an arithmetic progression with prime power modulus.</div></div><div><h3>Video</h3><div>For a video summary of this paper, please visit <span><span>https://youtu.be/Hq7jCPi1EjM</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 149-177"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190615","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":"Comparison of component groups of ℓ-adic and mod ℓ monodromy groups","authors":"Boyi Dai,&nbsp;Chun-Yin Hui","doi":"10.1016/j.jnt.2025.12.011","DOIUrl":"10.1016/j.jnt.2025.12.011","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>:</mo><msub><mrow><mi>Gal</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>→</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo><mo>}</mo></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> be a semisimple compatible system of <em>ℓ</em>-adic representations of a number field <em>K</em> that is arising from geometry. Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>⊂</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></msub></math></span> and <span><math><msub><mrow><mover><mrow><munder><mrow><mi>G</mi></mrow><mo>_</mo></munder></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>⊂</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></msub></math></span> be respectively the algebraic monodromy group and the full algebraic envelope of <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>. We prove that there is a natural isomorphism between the component groups <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><munder><mrow><mi>G</mi></mrow><mo>_</mo></munder></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> for all sufficiently large <em>ℓ</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 246-261"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190612","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":"Larsen's conjecture for elliptic curves over Q with analytic rank at most one","authors":"Seokhyun Choi,&nbsp;Bo-Hae Im","doi":"10.1016/j.jnt.2025.12.001","DOIUrl":"10.1016/j.jnt.2025.12.001","url":null,"abstract":"<div><div>We prove Larsen's conjecture for elliptic curves over <span><math><mi>Q</mi></math></span> with analytic rank at most 1. Specifically, let <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> be an elliptic curve over <span><math><mi>Q</mi></math></span>. If <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> has analytic rank at most 1, then we prove that for any topologically finitely generated subgroup <em>G</em> of <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover><mo>/</mo><mi>Q</mi><mo>)</mo></math></span>, the rank of <em>E</em> over the fixed subfield <span><math><msup><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msup></math></span> of <span><math><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></math></span> under <em>G</em> is infinite.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 1-14"},"PeriodicalIF":0.7,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039934","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}