Pub Date : 2025-08-21DOI: 10.1016/j.jnt.2025.07.014
Bogdan Nica
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.
我们计算沿三次曲线的单参数族的点的数目的方差。我们强调利用雅各布撒和的方差的显式评估。
{"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}
Pub Date : 2025-08-21DOI: 10.1016/j.jnt.2025.07.011
Qingfeng Sun , Qizhi Zhang
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.
{"title":"Mass distribution for holomorphic cusp forms on the vertical geodesic","authors":"Qingfeng Sun , 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}
Pub Date : 2025-08-21DOI: 10.1016/j.jnt.2025.07.018
Tapas Chatterjee , Sonam Garg
Murty and Saradha (2008) initiated a significant exploration into the transcendental nature of certain p-adic constants, focusing on the p-adic analogues of the Euler's constant and the Euler-Lehmer constant , where p is a rational prime with . Their work laid the foundation for understanding these constants in the context of p-adic analysis.
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 p-adic analogues of the Euler-Lehmer constants in this broader framework.
{"title":"Transcendental nature of p-adic Euler-Lehmer constants","authors":"Tapas Chatterjee , 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><</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}
Pub Date : 2025-08-21DOI: 10.1016/j.jnt.2025.07.017
Ingmar Metzler , Riccardo Zuffetti
Let L be an even indefinite lattice. We show that if L splits off a hyperbolic plane and a scaled hyperbolic plane, then the Kudla–Millson lift of genus 1 associated to L 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.
{"title":"Injectivity of the genus 1 Kudla–Millson lift on locally symmetric spaces","authors":"Ingmar Metzler , 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}
Pub Date : 2025-08-20DOI: 10.1016/j.jnt.2025.07.007
R. Balasubramanian , Sayan Dutta
We prove that if is a Sidon set so that , then where . 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 S and for any , we have for all but at most exceptions.
我们证明,如果一个= {a1,…,一个| |}⊂{1,2,…,n}是一个西顿,| | = n−L’,thenam = m⋅n + O (n7/8) + O (L1/2⋅n3/4), L = max{0,L '}。作为这一点的应用，我们给出了一些先前推导结果的简单证明。我们进一步证明了对于一个密集的西顿集S和对于任意ε>；0，对于所有n≤n但不超过Oε(N45+ε)的例外，∑a∈Sa=12n3/2+O（n11/8）。
{"title":"The m-th element of a Sidon set","authors":"R. Balasubramanian , 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>></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}
Pub Date : 2025-08-20DOI: 10.1016/j.jnt.2025.07.015
Takumi Anzawa
In this paper, we show that the cyclotomic symmetric multiple zeta values, independently proposed by Jarossay, Singar and Zhao, and Tasaka, span the space of the cyclotomic multiple zeta values modulo πi.
{"title":"Cyclotomic symmetric multiple zeta values span the space of cyclotomic multiple zeta values","authors":"Takumi Anzawa","doi":"10.1016/j.jnt.2025.07.015","DOIUrl":"10.1016/j.jnt.2025.07.015","url":null,"abstract":"<div><div>In this paper, we show that the cyclotomic symmetric multiple zeta values, independently proposed by Jarossay, Singar and Zhao, and Tasaka, span the space of the cyclotomic multiple zeta values modulo <em>πi</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 749-760"},"PeriodicalIF":0.7,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904484","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 : 2025-07-25DOI: 10.1016/j.jnt.2025.06.016
Joseph H. Silverman
We correct a constant appearing in an inequality, and explain how the change propagates through the paper to change various other constants. The revised result is the lower bound in which the fraction replaces the constant appearing in the original publication, and with the added requirement that .
{"title":"Corrigendum to: “A Lehmer-type lower bound for the canonical height on elliptic curves over function fields” [J. Number Theory 262 (2024) 506–538]","authors":"Joseph H. Silverman","doi":"10.1016/j.jnt.2025.06.016","DOIUrl":"10.1016/j.jnt.2025.06.016","url":null,"abstract":"<div><div>We correct a constant appearing in an inequality, and explain how the change propagates through the paper to change various other constants. The revised result is the lower bound<span><span><span><math><msub><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>E</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>18000</mn><mo>⋅</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>F</mi></mrow></msub><msup><mrow><mo>(</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mo>[</mo><mi>K</mi><mo>:</mo><mi>F</mi><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span> in which the fraction <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>18000</mn></mrow></mfrac></math></span> replaces the constant <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>10500</mn></mrow></mfrac></math></span> appearing in the original publication, and with the added requirement that <span><math><mo>[</mo><mi>K</mi><mo>:</mo><mi>F</mi><mo>]</mo><mo>≥</mo><mn>6</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 267-269"},"PeriodicalIF":0.6,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702882","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 : 2025-07-24DOI: 10.1016/j.jnt.2025.06.008
Yann Bugeaud , Bernard de Mathan
We give new examples of pairs composed of a real and a p-adic numbers that satisfy a conjecture on simultaneous multiplicative approximation by rational numbers formulated by Einsiedler and Kleinbock in 2007.
{"title":"Simultaneous multiplicative rational approximation to a real and a p-adic numbers","authors":"Yann Bugeaud , Bernard de Mathan","doi":"10.1016/j.jnt.2025.06.008","DOIUrl":"10.1016/j.jnt.2025.06.008","url":null,"abstract":"<div><div>We give new examples of pairs composed of a real and a <em>p</em>-adic numbers that satisfy a conjecture on simultaneous multiplicative approximation by rational numbers formulated by Einsiedler and Kleinbock in 2007.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 564-578"},"PeriodicalIF":0.7,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144749525","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 : 2025-07-22DOI: 10.1016/j.jnt.2025.06.012
C.G. Karthick Babu , E. Malavika , G.K. Viswanadham
We consider the Poissonian pair correlation of the sequence generated by the generalized polynomial , where p runs over the sequence of primes and α is an irrational number. We show that for any irrational of finite type, the sequence is not metric Poissonian. This is done by considering an additive problem similar to the even Goldbach conjecture. We also give upper and lower bounds for the additive energy of the sequence .
{"title":"Poissonian pair correlation of linear generalized monomials over primes","authors":"C.G. Karthick Babu , E. Malavika , G.K. Viswanadham","doi":"10.1016/j.jnt.2025.06.012","DOIUrl":"10.1016/j.jnt.2025.06.012","url":null,"abstract":"<div><div>We consider the Poissonian pair correlation of the sequence <span><math><msub><mrow><mo>(</mo><mo>⌊</mo><mi>p</mi><mi>α</mi><mo>⌋</mo><mo>)</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> generated by the generalized polynomial <span><math><mo>⌊</mo><mi>α</mi><mi>X</mi><mo>⌋</mo></math></span>, where <em>p</em> runs over the sequence of primes and <em>α</em> is an irrational number. We show that for any irrational <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> of finite type, the sequence <span><math><msub><mrow><mo>(</mo><mo>⌊</mo><mi>p</mi><mi>α</mi><mo>⌋</mo><mo>)</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span> is not metric Poissonian. This is done by considering an additive problem similar to the even Goldbach conjecture. We also give upper and lower bounds for the additive energy of the sequence <span><math><msub><mrow><mo>(</mo><mo>⌊</mo><mi>p</mi><mi>α</mi><mo>⌋</mo><mo>)</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 270-293"},"PeriodicalIF":0.6,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702881","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 : 2025-07-21DOI: 10.1016/j.jnt.2025.06.015
Youngmin Lee , Subong Lim
In this paper, we study an analogy of the heat operator to the skew-holomorphic Jacobi form case. Using this, we prove Bol's identity for skew-holomorphic Jacobi forms on . This induces a map from skew-holomorphic Jacobi forms of weight to those of weight . When , this map extends to skew-holomorphic harmonic Maass-Jacobi forms. In this case, we prove Zagier-type duality between Fourier coefficients of harmonic Maass-Jacobi forms and Fourier coefficients of weakly skew-holomorphic Jacobi forms.
{"title":"Bol's identity for skew-holomorphic Jacobi forms","authors":"Youngmin Lee , Subong Lim","doi":"10.1016/j.jnt.2025.06.015","DOIUrl":"10.1016/j.jnt.2025.06.015","url":null,"abstract":"<div><div>In this paper, we study an analogy of the heat operator to the skew-holomorphic Jacobi form case. Using this, we prove Bol's identity for skew-holomorphic Jacobi forms on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>j</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span>. This induces a map from skew-holomorphic Jacobi forms of weight <span><math><mo>−</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> to those of weight <span><math><mi>k</mi><mo>+</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mi>j</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mn>2</mn></math></span>. When <span><math><mi>n</mi><mo>=</mo><mi>j</mi><mo>=</mo><mn>1</mn></math></span>, this map extends to skew-holomorphic harmonic Maass-Jacobi forms. In this case, we prove Zagier-type duality between Fourier coefficients of harmonic Maass-Jacobi forms and Fourier coefficients of weakly skew-holomorphic Jacobi forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 216-237"},"PeriodicalIF":0.6,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702916","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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