实三次函数场除数类数的平均值

IF 1 4区 数学 Q1 MATHEMATICS
Yoonjin Lee, Jungyun Lee, Jinjoo Yoo
{"title":"实三次函数场除数类数的平均值","authors":"Yoonjin Lee, Jungyun Lee, Jinjoo Yoo","doi":"10.1515/math-2023-0160","DOIUrl":null,"url":null,"abstract":"We compute an asymptotic formula for the divisor class numbers of <jats:italic>real</jats:italic> cubic function fields <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>k</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mroot> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mroot> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{K}_{m}=k\\left(\\sqrt[3]{m})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{{\\mathbb{F}}}_{q}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a finite field with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>q</m:mi> </m:math> <jats:tex-math>q</jats:tex-math> </jats:alternatives> </jats:inline-formula> elements, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>q</m:mi> <m:mo>≡</m:mo> <m:mn>1</m:mn> <m:mspace width=\"0.3em\" /> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mi>mod</m:mi> </m:mrow> <m:mspace width=\"0.3em\" /> <m:mn>3</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>q\\equiv 1\\hspace{0.3em}\\left(\\mathrm{mod}\\hspace{0.3em}3)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> <m:mo>≔</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>k:= {{\\mathbb{F}}}_{q}\\left(T)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the rational function field, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>m\\in {{\\mathbb{F}}}_{q}\\left[T]</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a cube-free polynomial; in this case, the degree of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>m</m:mi> </m:math> <jats:tex-math>m</jats:tex-math> </jats:alternatives> </jats:inline-formula> is divisible by 3. For computation of our asymptotic formula, we find the average value of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>χ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>{| L\\left(s,\\chi )| }^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> evaluated at <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>s</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>s=1</jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_010.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>χ</m:mi> </m:math> <jats:tex-math>\\chi </jats:tex-math> </jats:alternatives> </jats:inline-formula> goes through the primitive cubic <jats:italic>even</jats:italic> Dirichlet characters of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_011.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>{{\\mathbb{F}}}_{q}\\left[T]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_012.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>χ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>L\\left(s,\\chi )</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the associated Dirichlet <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0160_eq_013.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>L</m:mi> </m:math> <jats:tex-math>L</jats:tex-math> </jats:alternatives> </jats:inline-formula>-function.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"20 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Average value of the divisor class numbers of real cubic function fields\",\"authors\":\"Yoonjin Lee, Jungyun Lee, Jinjoo Yoo\",\"doi\":\"10.1515/math-2023-0160\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We compute an asymptotic formula for the divisor class numbers of <jats:italic>real</jats:italic> cubic function fields <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi>k</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mroot> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mroot> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{K}_{m}=k\\\\left(\\\\sqrt[3]{m})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{{\\\\mathbb{F}}}_{q}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a finite field with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>q</m:mi> </m:math> <jats:tex-math>q</jats:tex-math> </jats:alternatives> </jats:inline-formula> elements, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_004.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>q</m:mi> <m:mo>≡</m:mo> <m:mn>1</m:mn> <m:mspace width=\\\"0.3em\\\" /> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mi>mod</m:mi> </m:mrow> <m:mspace width=\\\"0.3em\\\" /> <m:mn>3</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>q\\\\equiv 1\\\\hspace{0.3em}\\\\left(\\\\mathrm{mod}\\\\hspace{0.3em}3)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_005.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>k</m:mi> <m:mo>≔</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>k:= {{\\\\mathbb{F}}}_{q}\\\\left(T)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the rational function field, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_006.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>m\\\\in {{\\\\mathbb{F}}}_{q}\\\\left[T]</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a cube-free polynomial; in this case, the degree of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_007.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>m</m:mi> </m:math> <jats:tex-math>m</jats:tex-math> </jats:alternatives> </jats:inline-formula> is divisible by 3. For computation of our asymptotic formula, we find the average value of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_008.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>χ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>{| L\\\\left(s,\\\\chi )| }^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> evaluated at <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_009.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>s</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>s=1</jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_010.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>χ</m:mi> </m:math> <jats:tex-math>\\\\chi </jats:tex-math> </jats:alternatives> </jats:inline-formula> goes through the primitive cubic <jats:italic>even</jats:italic> Dirichlet characters of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_011.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>{{\\\\mathbb{F}}}_{q}\\\\left[T]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_012.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>χ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>L\\\\left(s,\\\\chi )</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the associated Dirichlet <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0160_eq_013.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>L</m:mi> </m:math> <jats:tex-math>L</jats:tex-math> </jats:alternatives> </jats:inline-formula>-function.\",\"PeriodicalId\":48713,\"journal\":{\"name\":\"Open Mathematics\",\"volume\":\"20 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-01-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Open Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/math-2023-0160\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/math-2023-0160","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

我们计算了实三次函数域 K m = k ( m 3 ) {K}_{m}=kleft(\sqrt[3]{m}) 的因子类数的渐近公式,其中 F q {{\mathbb{F}}}_{q} 是具有 q q 个元素的有限域,q ≡ 1 ( mod 3 ) q\equiv 1\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) , k ≔ F q ( T ) k:= {{\mathbb{F}}}_{q}\left(T) 是有理函数域,并且 m ∈ F q [ T ] m\in {{\mathbb{F}}}_{q}\left[T] 是无立方多项式;在这种情况下,m m 的阶数可以被 3 整除。为了计算渐近公式,我们要找到 ∣ L ( s , χ ) ∣ 2 {| L\left(s. \chi )| }^^ 的平均值、\当 χ \chi 经过 F q [ T ] {{\mathbb{F}}}_{q}\left[T] 的原始立方偶数 Dirichlet 字符时,在 s = 1 s=1 处求值,其中 L ( s , χ ) L\left(s,\chi ) 是相关的 Dirichlet L L - 函数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Average value of the divisor class numbers of real cubic function fields
We compute an asymptotic formula for the divisor class numbers of real cubic function fields K m = k ( m 3 ) {K}_{m}=k\left(\sqrt[3]{m}) , where F q {{\mathbb{F}}}_{q} is a finite field with q q elements, q 1 ( mod 3 ) q\equiv 1\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) , k F q ( T ) k:= {{\mathbb{F}}}_{q}\left(T) is the rational function field, and m F q [ T ] m\in {{\mathbb{F}}}_{q}\left[T] is a cube-free polynomial; in this case, the degree of m m is divisible by 3. For computation of our asymptotic formula, we find the average value of L ( s , χ ) 2 {| L\left(s,\chi )| }^{2} evaluated at s = 1 s=1 when χ \chi goes through the primitive cubic even Dirichlet characters of F q [ T ] {{\mathbb{F}}}_{q}\left[T] , where L ( s , χ ) L\left(s,\chi ) is the associated Dirichlet L L -function.
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来源期刊
Open Mathematics
Open Mathematics MATHEMATICS-
CiteScore
2.40
自引率
5.90%
发文量
67
审稿时长
16 weeks
期刊介绍: Open Mathematics - formerly Central European Journal of Mathematics Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant, original and relevant works in all areas of mathematics. The journal provides the readers with free, instant, and permanent access to all content worldwide; and the authors with extensive promotion of published articles, long-time preservation, language-correction services, no space constraints and immediate publication. Open Mathematics is listed in Thomson Reuters - Current Contents/Physical, Chemical and Earth Sciences. Our standard policy requires each paper to be reviewed by at least two Referees and the peer-review process is single-blind. Aims and Scope The journal aims at presenting high-impact and relevant research on topics across the full span of mathematics. Coverage includes:
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