On the number of subgroups of the group $\mathbb{Z}_{m_{1}} \times \mathbb{Z}_{m_{2}}$ with $m_{1}m_{2}\leq x$ such that $m_{1}m_{2}$ is a $k$-th power
Yankun Sui, Dan Liu, Boling Zhou
TL;DR
We study the problem of counting subgroups of $Z_{m_1} \times Z_{m_2}$ with $m_1 m_2 \le x$ and $m_1 m_2$ a $k$-th power. The authors develop the Dirichlet series $F(s;k)$ for the restricted sum and factorize it as $F(s;2)=\zeta(2s-1)H(s;2)$ and $F(s;3)=\zeta^4(3s-1)\zeta(6s-3)H(s;3)$ with $H(s;k)$ analytic for $\Re s>1/2$, enabling analytic continuation and contour integration. Using Perron's formula and residue calculus, they derive sharp asymptotics: $T_2(x)=c_2 x + O(x^{19/28+\varepsilon})$ and $T_3(x)=x^{2/3}\sum_{j=0}^{4} a_j \log^j x + O(x^{5/8+\varepsilon})$, where the constants $c_2$ and $a_j$ are explicit. The results advance understanding of average subgroup counts under a $k$-th power constraint and illustrate a robust Dirichlet-series/contour-integration approach to subgroup-count problems in restricted regions.
Abstract
Let ${\Bbb Z}_{m}$ be the additive group of residue classes modulo $m$ and $s(m_{1},m_{2})$ denote the number of subgroups of the group ${\Bbb Z}_{m_{1}}\times {\Bbb Z}_{m_{2}}$, where $m_{1}$ and $m_{2}$ are arbitrary positive integers. We consider sums of type $\sum\limits_{\substack{m_{1}m_{2}\leq x \\ m_{1}m_{2}\in N_{k}}}s(m_{1},m_{2})$, where $N_{k}$ is the set of $k$-th power of natural numbers. In particular, we deduce asymptotic formulas with $k=2$ and $k=3$.