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Continuous lower bounds for moments of the mixed product of twisted L-functions
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.003
Guohua Chen , Weiping Li , Tianze Wang
{"title":"Continuous lower bounds for moments of the mixed product of twisted L-functions","authors":"Guohua Chen ,&nbsp;Weiping Li ,&nbsp;Tianze Wang","doi":"10.1016/j.jnt.2024.12.003","DOIUrl":"10.1016/j.jnt.2024.12.003","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> be two distinct cuspidal holomorphic Hecke eigenforms of even weight <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>κ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> for the modular group <span><math><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> respectively. Let <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><mi>χ</mi><mo>)</mo><mo>,</mo><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊗</mo><mi>χ</mi><mo>)</mo></math></span> denote twisted <em>L</em>-functions associated to <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> twisted by a primitive Dirichlet character <em>χ</em> modulo <em>q</em> respectively. In this paper, we obtain sharp lower bounds for all positive real <em>k</em>-th <span><math><mo>(</mo><mi>k</mi><mo>&gt;</mo><mn>1</mn><mo>)</mo></math></span> moments of the mixed product of these two twisted <em>L</em>-functions at the central value, which extend the previous results.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 438-457"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453738","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}
引用次数: 0
On the distribution of Ω(n)−ω(n)
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.002
Biao Wang
{"title":"On the distribution of Ω(n)−ω(n)","authors":"Biao Wang","doi":"10.1016/j.jnt.2024.12.002","DOIUrl":"10.1016/j.jnt.2024.12.002","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of all prime factors and distinct prime factors of <em>n</em>, respectively. In 1955, Rényi found the density for the numbers <em>n</em> such that <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi></math></span> for all integers <span><math><mi>k</mi><mo>≥</mo><mn>0</mn></math></span>. In this paper, we generalize Rényi's theorem and give a short and elementary proof. Moreover, we show that the distribution of <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> displays a disjoint form with the arithmetic functions of invariant averages under multiplications. As a consequence, we obtain some ergodic theorems on <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> that build connections among Rényi's theorem, the prime number theorem, Bergelson-Richter's theorem and Loyd's theorem.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 423-437"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453737","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}
引用次数: 0
On the Rankin-Selberg gamma factor of simple supercuspidal representations of the even unitary group for p-adic local fields
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.015
Philip Barron, Yu Xin
{"title":"On the Rankin-Selberg gamma factor of simple supercuspidal representations of the even unitary group for p-adic local fields","authors":"Philip Barron,&nbsp;Yu Xin","doi":"10.1016/j.jnt.2025.01.015","DOIUrl":"10.1016/j.jnt.2025.01.015","url":null,"abstract":"<div><div>Let <em>π</em> be a simple supercuspidal representation of the quasi-split unramified even unitary group with respect to an unramified quadratic extension <span><math><mi>E</mi><mo>/</mo><mi>F</mi></math></span> of <em>p</em>-adic fields. We compute the Rankin-Selberg gamma factor for rank-1 twists of <em>π</em> by a tamely ramified character of <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span>. For non-dyadic cases, the gamma factor can also be derived from the fact that endoscopic lift is also simple supercuspidal. For the dyadic case, the result is original. We expect to extend the result on the endoscopic lift to the dyadic case with our computation.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 526-547"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479480","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}
引用次数: 0
On the Kodaira types of elliptic curves with potentially good supersingular reduction
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.008
Haiyang Wang
{"title":"On the Kodaira types of elliptic curves with potentially good supersingular reduction","authors":"Haiyang Wang","doi":"10.1016/j.jnt.2025.01.008","DOIUrl":"10.1016/j.jnt.2025.01.008","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> be a Henselian discrete valuation domain with field of fractions <em>K</em>. Assume that <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> has algebraically closed residue field <em>k</em>. Let <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> such that the base change <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>/</mo><mi>L</mi></math></span> has semi-stable reduction.</div><div>It is natural to wonder whether specific properties of the semi-stable reduction and of the extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> impose restrictions on what types of Kodaira type the special fiber of <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> may have. In this paper we study the restrictions imposed on the reduction type when the extension <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> is wildly ramified of degree 2, and the curve <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> has potentially good supersingular reduction. We also analyze the possible reduction types of two isogenous elliptic curves with these properties.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 283-307"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445482","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}
引用次数: 0
Sums of four polygonal numbers: Precise formulas
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.016
Jialin Li, Haowu Wang
{"title":"Sums of four polygonal numbers: Precise formulas","authors":"Jialin Li,&nbsp;Haowu Wang","doi":"10.1016/j.jnt.2025.01.016","DOIUrl":"10.1016/j.jnt.2025.01.016","url":null,"abstract":"<div><div>In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized <em>m</em>-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are given as <span><math><mi>Z</mi></math></span>-linear combinations of Hurwitz class numbers. As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of <span><math><mi>ϑ</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></math></span>, <span><math><mi>η</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mrow><mn>12</mn></mrow></msup></math></span>, <span><math><mi>η</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></math></span> and <span><math><mi>η</mi><msup><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mrow><mn>8</mn></mrow></msup><mi>η</mi><msup><mrow><mo>(</mo><mn>2</mn><mi>τ</mi><mo>)</mo></mrow><mrow><mn>8</mn></mrow></msup></math></span> in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 407-422"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445487","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}
引用次数: 0
Arithmetic of cubic number fields: Jacobi–Perron, Pythagoras, and indecomposables
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.001
Vítězslav Kala , Ester Sgallová , Magdaléna Tinková
{"title":"Arithmetic of cubic number fields: Jacobi–Perron, Pythagoras, and indecomposables","authors":"Vítězslav Kala ,&nbsp;Ester Sgallová ,&nbsp;Magdaléna Tinková","doi":"10.1016/j.jnt.2024.12.001","DOIUrl":"10.1016/j.jnt.2024.12.001","url":null,"abstract":"<div><div>We study a new connection between multidimensional continued fractions, such as Jacobi–Perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposables to formulate our conclusions.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"273 ","pages":"Pages 37-95"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487941","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}
引用次数: 0
Zeros of Dirichlet L-functions on the critical line
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2024.12.005
Keiju Sono
{"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}
引用次数: 0
p-adic equidistribution of modular geodesics and of CM points on Shimura curves
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.010
Patricio Pérez-Piña
{"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}
引用次数: 0
Modular symbols and equivariant birational invariants
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.006
Zhijia Zhang
{"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}
引用次数: 0
Averages of multiplicative functions along equidistributed sequences
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-02-17 DOI: 10.1016/j.jnt.2025.01.005
Stephanie Chan , Peter Koymans , Carlo Pagano , Efthymios Sofos
{"title":"Averages of multiplicative functions along equidistributed sequences","authors":"Stephanie Chan ,&nbsp;Peter Koymans ,&nbsp;Carlo Pagano ,&nbsp;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}
引用次数: 0
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