论文标题
在$ {\ Mathbb {z}} _ {m} \ times {\ Mathbb {z}} _ {n} $带有$ Mn \ leq x $的$ {\ Mathbb {z}} _ {m}} _ {m}} _ {m}} _ {\ times {\ times {\ times {\ times {\ times {\ time
On the weighted average number of subgroups of ${\mathbb {Z}}_{m}\times {\mathbb {Z}}_{n}$ with $mn\leq x$
论文作者
论文摘要
令$ \ mathbb {z} _ {m} $为残基类的加法组modulo $ m $。对于任何积极的整数$ m $和$ n $，让$ s（m，n）$和$ c（m，n）$表示组$ {\ mathbb {z}} _ {m} _ {m} _ {m} \ times {\ times {\ mathbb {\ sathbb {z} _} $的子组和周期子组总数。定义$ \ widetilde {d} _ {s}（x）= \ sum_ {mn \ leq x} s（m，n）\ log \ frac {x} {x} {mn} {mn} \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ wideteDeLe {d} x} c（m，n）\ log \ frac {x} {mn}。 $$在本文中，我们研究功能的渐近行为$ \ widetilde {d} _ {s}（x）$和$ \ widetilde {d} _ {c}（x）$。
Let $\mathbb{Z}_{m}$ be the additive group of residue classes modulo $m$. For any positive integers $m$ and $n$, let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and cyclic subgroups of the group ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{n}$, respectively. Define $$ \widetilde{D}_{s}(x) = \sum_{mn\leq x}s(m,n)\log\frac{x}{mn} \quad \quad \widetilde{D}_{c}(x) = \sum_{mn\leq x}c(m,n)\log\frac{x}{mn}. $$ In this paper, we study the asymptotic behaviour of functions $\widetilde{D}_{s}(x)$ and $\widetilde{D}_{c}(x)$.
