算术函数加权平均数的拉普拉斯卷积

IF 1 3区 数学 Q1 MATHEMATICS Forum Mathematicum Pub Date : 2024-04-23 DOI:10.1515/forum-2023-0259
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Laplace convolutions of weighted averages of arithmetical functions
Let G ( g ; x ) := n x g ( 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 g j ( n ) , j { 1 , , N } {g_{j}(n),\,j\in\{1,\dots,N\}} as an integral involving the convolution (in the sense of Laplace) of G j ( 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.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Forum Mathematicum
Forum Mathematicum 数学-数学
CiteScore
1.60
自引率
0.00%
发文量
78
审稿时长
6-12 weeks
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.
期刊最新文献
Is addition definable from multiplication and successor? The stable category of monomorphisms between (Gorenstein) projective modules with applications Big pure projective modules over commutative noetherian rings: Comparison with the completion Discrete Ω-results for the Riemann zeta function Any Sasakian structure is approximated by embeddings into spheres
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1