{"title":"Zeros of Dirichlet L-functions on the critical line","authors":"Keiju Sono","doi":"10.1016/j.jnt.2024.12.005","DOIUrl":"10.1016/j.jnt.2024.12.005","url":null,"abstract":"<div><div>In this paper, we estimate the proportion of zeros of Dirichlet <em>L</em>-functions on the critical line. Using Feng's mollifier <span><span>[8]</span></span> and an asymptotic formula for the mean square of Dirichlet <em>L</em>-functions introduced in <span><span>[7]</span></span>, we prove that, averaged over primitive characters and conductors, at least 61.07% of the zeros of Dirichlet <em>L</em>-functions lie on the critical line, and at least 60.44% of the zeros are simple and lie on the critical line. These results improve upon the work of Conrey, Iwaniec, and Soundararajan in <span><span>[6]</span></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 348-388"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445485","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":"p-adic equidistribution of modular geodesics and of CM points on Shimura curves","authors":"Patricio Pérez-Piña","doi":"10.1016/j.jnt.2025.01.010","DOIUrl":"10.1016/j.jnt.2025.01.010","url":null,"abstract":"<div><div>We propose a <em>p</em>-adic version of Duke's Theorem on the equidistribution of closed geodesics on modular curves. Our approach concerns quadratic fields split at <em>p</em> as well as a <em>p</em>-adic covering of the modular curve. We also prove an equidistribution result of CM points in the <em>p</em>-adic space attached to Shimura curves.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 259-282"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437074","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 symbols and equivariant birational invariants","authors":"Zhijia Zhang","doi":"10.1016/j.jnt.2025.01.006","DOIUrl":"10.1016/j.jnt.2025.01.006","url":null,"abstract":"<div><div>We study relations between the classical modular symbols associated with congruence subgroups and Kontsevich-Pestun-Tschinkel groups <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> associated with finite abelian groups <em>G</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 308-327"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445483","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":"Pisot numbers, Salem numbers, and generalised polynomials","authors":"Jakub Byszewski , Jakub Konieczny","doi":"10.1016/j.jnt.2025.01.001","DOIUrl":"10.1016/j.jnt.2025.01.001","url":null,"abstract":"<div><div>We study sets of integers that can be defined by the vanishing of a generalised polynomial expression. We show that this includes sets of values of linear recurrent sequences of Salem type and some linear recurrent sequences of Pisot type. To this end, we introduce the notion of a generalised polynomial on a number field. We establish a connection between the existence of generalised polynomial expressions for sets of values of linear recurrent sequences and for subsemigroups of multiplicative groups of number fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 475-503"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stephanie Chan , Peter Koymans , Carlo Pagano , Efthymios Sofos
{"title":"Averages of multiplicative functions along equidistributed sequences","authors":"Stephanie Chan , Peter Koymans , Carlo Pagano , Efthymios Sofos","doi":"10.1016/j.jnt.2025.01.005","DOIUrl":"10.1016/j.jnt.2025.01.005","url":null,"abstract":"<div><div>For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"273 ","pages":"Pages 1-36"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Counting numbers that are divisible by the product of their digits","authors":"Qizheng He , Carlo Sanna","doi":"10.1016/j.jnt.2025.01.002","DOIUrl":"10.1016/j.jnt.2025.01.002","url":null,"abstract":"<div><div>Let <span><math><mi>b</mi><mo>≥</mo><mn>3</mn></math></span> be a positive integer. A natural number is said to be a <em>base-b Zuckerman number</em> if it is divisible by the product of its base-<em>b</em> digits (which consequently must be all nonzero). Let <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be the set of base-<em>b</em> Zuckerman numbers that do not exceed <em>x</em>, and assume that <span><math><mi>x</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span>.</div><div>First, we prove an upper bound of the form <span><math><mo>|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo><</mo><msup><mrow><mi>x</mi></mrow><mrow><msubsup><mrow><mi>z</mi></mrow><mrow><mi>b</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, where <span><math><msubsup><mrow><mi>z</mi></mrow><mrow><mi>b</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> is an effectively computable constant. In particular, we have that <span><math><msubsup><mrow><mi>z</mi></mrow><mrow><mn>10</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo>=</mo><mn>0.665</mn><mo>…</mo></math></span>, which improves upon the previous upper bound <span><math><mo>|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo><</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>0.717</mn></mrow></msup></math></span> due to Sanna. Moreover, we prove that <span><math><mo>|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>></mo><msup><mrow><mi>x</mi></mrow><mrow><mn>0.204</mn></mrow></msup></math></span>, which improves upon the previous lower bound <span><math><mo>|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>></mo><msup><mrow><mi>x</mi></mrow><mrow><mn>0.122</mn></mrow></msup></math></span>, due to De Koninck and Luca.</div><div>Second, we provide a heuristic suggesting that <span><math><mo>|</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, where <span><math><msub><mrow><mi>z</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> is an effectively computable constant. In particular, we have that <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>=</mo><mn>0.419</mn><mo>…</mo></math></span>.</div><div>Third, we provide algorithms to count, respectively enumerate, the elements of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>b</mi></mrow></msub><m","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 34-59"},"PeriodicalIF":0.6,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143478632","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":"Iwasawa Theory of elliptic curves at supersingular primes over higher rank Iwasawa extensions","authors":"Byoung Du (BD) Kim","doi":"10.1016/j.jnt.2024.11.005","DOIUrl":"10.1016/j.jnt.2024.11.005","url":null,"abstract":"<div><div>Suppose <em>K</em> is an imaginary quadratic field over which the prime <em>p</em> is inert, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> is its Iwasawa extension of rank 2, and <em>E</em> is an elliptic curve defined over <em>K</em> with good supersingular reduction at the prime above <em>p</em>. Unlike the case where <em>p</em> splits completely over <em>K</em> as in the author's previous work, no good Iwasawa Theory has been established in this case. We construct series of local points of <em>E</em> over <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfying certain norm relations by Fontaine's theory of group schemes, establish the algebraic side of Iwasawa Theory in this case, compatible with the author's theory on the analytic side, and propose a conjecture relating the algebraic side of the theory and the analytic side of it. (And, to do that, we also show that the author's previous work on the analytic side, which the author did only for the primes that split over <em>K</em>, also applies to the inert primes.)</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 189-215"},"PeriodicalIF":0.6,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global theta lifting and automorphic periods associated to nilpotent orbits","authors":"Bryan Peng Jun Wang","doi":"10.1016/j.jnt.2024.11.004","DOIUrl":"10.1016/j.jnt.2024.11.004","url":null,"abstract":"<div><div>A systematic way to organise the interesting periods of automorphic forms on a reductive group <em>G</em> is via the theory of nilpotent orbits of <em>G</em>. On the other hand, it is known that the theta correspondence can be used effectively to relate automorphic periods on each member of a dual pair. In this paper, we establish this relation in full generality, facilitated by a certain transfer of nilpotent orbits via moment maps. This is the analogous global result to the local result previously established by Gomez and Zhu.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 122-149"},"PeriodicalIF":0.6,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128937","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 number of points with bounded dynamical canonical height","authors":"Kohei Takehira","doi":"10.1016/j.jnt.2024.11.007","DOIUrl":"10.1016/j.jnt.2024.11.007","url":null,"abstract":"<div><div>This paper discusses the number of points for which the dynamical canonical height is less than or equal to a given value. The height function is a fundamental and important tool in number theory to capture the “number-theoretic complexity” of a point. Asymptotic formulas for the number of points in projective space below a given height have been studied by Schanuel <span><span>[Sch64]</span></span>, for example, and their coefficients can be written by class numbers, regulators, special values of the Dedekind zeta function, and other number theoretically interesting values. We consider an analogous problem for dynamical canonical height, a dynamical analogue of the height function in number theory, introduced by Call and Silverman <span><span>[CS93]</span></span>. The main tool of this study is the dynamical height zeta function studied by Hsia <span><span>[Hsi97]</span></span>. In this paper, we give explicit formulas for the dynamical height zeta function in special cases, derive general formulas for obtaining asymptotic behavior from certain functions, and combine them to derive asymptotic behavior for the number of points with bounded dynamical canonical height.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 216-245"},"PeriodicalIF":0.6,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128945","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":"Explicit cocycle of the Dedekind-Rademacher cohomology class and the Darmon-Dasgupta measures","authors":"Jae Hyung Sim","doi":"10.1016/j.jnt.2024.11.006","DOIUrl":"10.1016/j.jnt.2024.11.006","url":null,"abstract":"<div><div>The work of Darmon, Pozzi, and Vonk <span><span>[3]</span></span> has recently shown that the RM-values of the Dedekind-Rademacher cocycle <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>D</mi><mi>R</mi></mrow></msub></math></span> are Gross-Stark units up to controlled torsion. The authors of <span><span>[3]</span></span> remarked that the measure-valued cohomology class, which we denote <span><math><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>D</mi><mi>R</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>, underlying <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>D</mi><mi>R</mi></mrow></msub></math></span> is the level 1 incarnation of earlier constructions in <span><span>[1]</span></span>. In this paper, we make this relationship explicit by computing a concrete cocycle representative of an adelic incarnation <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>D</mi><mi>R</mi></mrow></msub></math></span> by tracing the construction of the cohomology class and comparing periods of weight 2 Eisenstein series. While maintaining a global perspective in our computations, we configure the appropriate method of smoothing cocycles which exactly yields the <em>p</em>-adic measures of <span><span>[1]</span></span> when applied to <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>D</mi><mi>R</mi></mrow></msub></math></span>. These methods will also explain the optional degree zero condition imposed in <span><span>[1]</span></span> which was remarked upon in <span><span>[6]</span></span> and <span><span>[7]</span></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 150-188"},"PeriodicalIF":0.6,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128938","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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