Abstract

Let s(m,n) and c(m,n) denote the total number of subgroups and cyclic subgroups of Zm×Zn, respectively. For any x≥1, we consider the asymptotic behavior of Ds(x):=∑mn≤xs(m,n) and Dc(x):=∑mn≤xc(m,n) and obtain two asymptotic formulas by using the method of exponential sums. We also study the upper bound of the mean-square estimate of Δs(x) and Δc(x).

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