AI 中文总结
研究群${\Bbb Z}_{n_1}\times {\Bbb Z}_{n_2}$子群数量和循环子群数量的和式渐近行为，给出更精确渐近结果并研究误差项均值。
AI 中文摘要
设${\mathbb Z}_{n}$为模$n$的剩余类加法群。对于任意正整数$n_1$和$n_2$，分别用$s(n_1,n_2)$和$c(n_1,n_2)$表示群${\mathbb Z}_{n_1}\times {\mathbb Z}_{n_2}$的子群数量和循环子群数量。本文旨在研究和式$\sum_{n_1,n_2\le x}s(n_1,n_2)$与$\sum_{n_1,n_2\le x}c(n_1,n_2)$的渐近行为，给出了这些和式更精确的渐近结果，并研究了误差项的均值。
英文摘要
Let ${\mathbb Z}_{n}$ be the additive group of residue classes modulo $n.$ For any positive integers $n_1$ and $n_2$, let $s(n_1,n_2)$ and $c(n_1,n_2)$ denote the number of subgroups and the number of cyclic subgroups of the group ${\mathbb Z}_{n_1}\times {\mathbb Z}_{n_2}$, respectively. The aim of this paper is to study the asymptotic behavior of the sums $\sum_{n_1,n_2\le x}s(n_1,n_2)$ and $\sum_{n_1,n_2\le x}c(n_1,n_2)$. Some sharper asymptotic results are given for the these sums. Mean values of the error terms are also studied.
