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http://dx.doi.org/10.4134/JKMS.j180828

COUNTING SUBRINGS OF THE RING ℤm × ℤn  

Toth, Laszlo (Department of Mathematics University of Pecs)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.6, 2019 , pp. 1599-1611 More about this Journal
Abstract
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 $(m,n){\mapsto}N^{u,s}(m,n)$ 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 $\sum_{m,n{\leq}x}N^{(s)}(m,n)$, 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;
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