Number of subgroups of $\mathbb Z _m \times \mathbb Z_n$

Question

Let $\mathbb Z_m$ denote the additive group of residue classes modulo $m$.

Is there a closed form for the number of subgroups of $\mathbb Z_m\times\mathbb Z_n$?

Answer 1

There is a closed form if you allow for sums: According to Theorem 3 from the paper on the subgroups of the group $\mathbb Z_m\times\mathbb Z_n$ by Mario Hampejs, Nicki Holighaus, László Tóth and Christoph Wiesmeyr, the number of subgroups of $\mathbb Z_m\times\mathbb Z_n$ is $$\sum_{a|m\text{ and } b|n} \gcd(a,b)=\sum_{d|\gcd(m,n)} \phi(d)\tau\left(\frac md\right)\tau\left(\frac nd\right)=\sum_{d|\gcd(m,n)} d\cdot\tau\left(\frac{mn}{d^2}\right),$$

where $\phi$ is Euler's totient function and $\tau(n)$ is the number of positive divisors of $n$.