Counting subrings of the ring $\mathbb Z_m \times \mathbb Z_n$
J. Korean Math. Soc. 2019 Vol. 56, No. 6, 1599-1611
https://doi.org/10.4134/JKMS.j180828
Published online November 1, 2019
L\'aszl\'o T\'oth
University of P\'ecs
Abstract : 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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