J. Korean Math. Soc. 2019; 56(6): 1599-1611
Online first article July 17, 2019 Printed November 1, 2019
https://doi.org/10.4134/JKMS.j180828
Copyright © The Korean Mathematical Society.
L\'aszl\'o T\'oth
University of P\'ecs
Let $m,n\in \N$. We represent the additive subgroups of the ring $\Z_m \times \Z_n$, 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 $\Z_m \times \Z_n$ 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.
Keywords: subgroup, subring, ideal, number of subrings, multiplicative arithmetic function of two variables, asymptotic formula, Dirichlet divisor problem
MSC numbers: Primary 11N45, 20K27; Secondary 11A25, 13A99
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