{"title":"一些完全乘法算术函数的多重狄利克雷级数","authors":"Nabil Tahmi, Abdallah Derbal","doi":"10.7546/nntdm.2022.28.4.603-616","DOIUrl":null,"url":null,"abstract":"Let f_r: \\mathbb{N}^r \\longrightarrow \\mathbb{C} be an arithmetic function of r variables, where r\\geq 2. We study multiple Dirichlet series defined by \\begin{equation*} D(f_r,s_1,\\ldots,s_r)=\\sum\\limits_{\\substack{n_1,\\ldots,n_r=1 \\\\ (n_1,\\ldots,n_r)=1}}^{+\\infty}\\frac{f_r(n_1,\\ldots,n_r)}{n_1^{s_1}\\cdots n_r^{s_r}}, \\end{equation*} where f_r(n_1,\\ldots,n_r)=f(n_1)\\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Some multiple Dirichlet series of completely multiplicative arithmetic functions\",\"authors\":\"Nabil Tahmi, Abdallah Derbal\",\"doi\":\"10.7546/nntdm.2022.28.4.603-616\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let f_r: \\\\mathbb{N}^r \\\\longrightarrow \\\\mathbb{C} be an arithmetic function of r variables, where r\\\\geq 2. We study multiple Dirichlet series defined by \\\\begin{equation*} D(f_r,s_1,\\\\ldots,s_r)=\\\\sum\\\\limits_{\\\\substack{n_1,\\\\ldots,n_r=1 \\\\\\\\ (n_1,\\\\ldots,n_r)=1}}^{+\\\\infty}\\\\frac{f_r(n_1,\\\\ldots,n_r)}{n_1^{s_1}\\\\cdots n_r^{s_r}}, \\\\end{equation*} where f_r(n_1,\\\\ldots,n_r)=f(n_1)\\\\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.\",\"PeriodicalId\":44060,\"journal\":{\"name\":\"Notes on Number Theory and Discrete Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-10-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Notes on Number Theory and Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7546/nntdm.2022.28.4.603-616\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Notes on Number Theory and Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7546/nntdm.2022.28.4.603-616","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Some multiple Dirichlet series of completely multiplicative arithmetic functions
Let f_r: \mathbb{N}^r \longrightarrow \mathbb{C} be an arithmetic function of r variables, where r\geq 2. We study multiple Dirichlet series defined by \begin{equation*} D(f_r,s_1,\ldots,s_r)=\sum\limits_{\substack{n_1,\ldots,n_r=1 \\ (n_1,\ldots,n_r)=1}}^{+\infty}\frac{f_r(n_1,\ldots,n_r)}{n_1^{s_1}\cdots n_r^{s_r}}, \end{equation*} where f_r(n_1,\ldots,n_r)=f(n_1)\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.