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

Let $m,n\in \mathbb{N}$. We represent the additive subgroups of the ring ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$, 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 ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$ 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.