Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini
{"title":"算术函数加权平均数的拉普拉斯卷积","authors":"Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini","doi":"10.1515/forum-2023-0259","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop=\"true\" symmetric=\"true\">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0246.png\" /> <jats:tex-math>{G(g;x):=\\sum_{n\\leq x}g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the summatory function of an arithmetical function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0403.png\" /> <jats:tex-math>{g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we prove that we can write weighted averages of an arbitrary fixed number <jats:italic>N</jats:italic> of arithmetical functions <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0414.png\" /> <jats:tex-math>{g_{j}(n),\\,j\\in\\{1,\\dots,N\\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as an integral involving the convolution (in the sense of Laplace) of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0257.png\" /> <jats:tex-math>{G_{j}(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0424.png\" /> <jats:tex-math>{j\\in\\{1,\\dots,N\\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"105 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Laplace convolutions of weighted averages of arithmetical functions\",\"authors\":\"Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini\",\"doi\":\"10.1515/forum-2023-0259\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop=\\\"true\\\" symmetric=\\\"true\\\">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0259_eq_0246.png\\\" /> <jats:tex-math>{G(g;x):=\\\\sum_{n\\\\leq x}g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the summatory function of an arithmetical function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0259_eq_0403.png\\\" /> <jats:tex-math>{g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we prove that we can write weighted averages of an arbitrary fixed number <jats:italic>N</jats:italic> of arithmetical functions <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\\\"4.2pt\\\">,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\\\"normal\\\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\\\"false\\\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0259_eq_0414.png\\\" /> <jats:tex-math>{g_{j}(n),\\\\,j\\\\in\\\\{1,\\\\dots,N\\\\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as an integral involving the convolution (in the sense of Laplace) of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0259_eq_0257.png\\\" /> <jats:tex-math>{G_{j}(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\\\"normal\\\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\\\"false\\\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0259_eq_0424.png\\\" /> <jats:tex-math>{j\\\\in\\\\{1,\\\\dots,N\\\\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.\",\"PeriodicalId\":12433,\"journal\":{\"name\":\"Forum Mathematicum\",\"volume\":\"105 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/forum-2023-0259\",\"RegionNum\":3,\"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":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0259","RegionNum":3,"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
设 G ( g ; x ) := ∑ n ≤ x g ( n ) {G(g;x):=\sum_{n\leq x}g(n)} 为算术函数 g ( n ) {g(n)} 的求和函数。本文将证明,我们可以写出任意固定数量 N 的算术函数 g j ( n ) , j ∈ { 1 , ... , N } 的加权平均数 {g_{j}(n),\,j\in\{1,\dots,N\}} 是一个涉及 G j ( x ) {G_{j}(x)} 的卷积(拉普拉斯意义上)的积分,j∈ { 1 , ... , N }。 {j\in\{1,\dots,N\}} . .此外,我们还证明了一个特性,它使我们能够以非常简单自然的方式获得关于算术函数平均数的已知结果,并克服了一些著名问题的技术限制。
Laplace convolutions of weighted averages of arithmetical functions
Let G(g;x):=∑n≤xg(n){G(g;x):=\sum_{n\leq x}g(n)} be the summatory function of an arithmetical function g(n){g(n)}. In this paper, we prove that we can write weighted averages of an arbitrary fixed number N of arithmetical functions gj(n),j∈{1,…,N}{g_{j}(n),\,j\in\{1,\dots,N\}} as an integral involving the convolution (in the sense of Laplace) of Gj(x){G_{j}(x)}, j∈{1,…,N}{j\in\{1,\dots,N\}}. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.
期刊介绍:
Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.