{"title":"Counting subrings of the ring $\\mathbb Z_m \\times \\mathbb Z_n$","authors":"L. Tóth","doi":"10.4134/JKMS.j180828","DOIUrl":null,"url":null,"abstract":"Let $m,n\\in \\Bbb{N}$. We represent the additive subgroups of the ring $\\Bbb{Z}_m \\times \\Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\\Bbb{Z}_m \\times \\Bbb{Z}_n$ and its unital subrings, respectively. We show that the functions $(m,n)\\mapsto N^{(s)}(m,n)$ and $(m,n)\\mapsto N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\\sum_{m,n\\le x} N^{(s)}(m,n)$, the error term of which is closely related to the Dirichlet divisor problem.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2018-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Korean Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4134/JKMS.j180828","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$ and its unital subrings, respectively. We show that the functions $(m,n)\mapsto N^{(s)}(m,n)$ and $(m,n)\mapsto N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\sum_{m,n\le x} N^{(s)}(m,n)$, the error term of which is closely related to the Dirichlet divisor problem.
期刊介绍:
This journal endeavors to publish significant research of broad interests in pure and applied mathematics. One volume is published each year, and each volume consists of six issues (January, March, May, July, September, November).