On the average number of subgroups of the group \Bbb Z_n1× \Bbb Z_n2

2026/07/19 by Zhai Wenguang
Mathematics · #math.NT

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Abstract

Let \mathbb Z_n be the additive group of residue classes modulo n. For any positive integers n₁ and n₂, let s(n₁,n₂) and c(n₁,n₂) denote the number of subgroups and the number of cyclic subgroups of the group \mathbb Z_n1× \mathbb Z_n2, respectively. The aim of this paper is to study the asymptotic behavior of the sums ∑_{n1,n2≤ x}s(n₁,n₂) and ∑_{n1,n2≤ x}c(n₁,n₂). Some sharper asymptotic results are given for the these sums. Mean values of the error terms are also studied.

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