Counting subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$

TL;DR

This paper provides explicit formulas and properties for counting subrings and unital subrings of the ring a5_m imes a5_n, including multiplicativity, Dirichlet series, and asymptotic behavior.

Contribution

It introduces explicit formulas for the number of subrings and unital subrings of a5_m imes a5_n, and analyzes their multiplicative and asymptotic properties.

Findings

Functions are multiplicative in two variables.

Dirichlet series are expressed via the Riemann zeta function.

Asymptotic formula relates to the Dirichlet divisor problem.

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.