互素阶有限循环群直积自同构计数函数的分布

The Distribution of the Automorphism Counting Function of the Direct Product of Finite Cyclic Groups of Coprime Order

摘要: 设$\varphi \left(m,n\right)$ 为两个有限循环群的直积${\mathbb{Z}}_m\times {\mathbb{Z}}_n$ 的自同构的个数，本文考虑了当$m$ 、$n$ 互素时$\varphi \left(m,n\right)$ 的分布，即$\left(m,n\right)=1$ 时满足$\varphi \left(m,n\right)\le x$ 的$\left(m,n\right)$ 有多少对，利用双曲求和法、卷积法等解析数论方法得到了$\varphi \left(m,n\right)$ 分布的渐进表达式。

Abstract: Let $\varphi \left(m,n\right)$ be the number of automorphisms of the direct product of two finite cyclic groups ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$ . This paper considers the distribution of $\varphi \left(m,n\right)$ when $m$ and $n$ are coprime, that is, how many pairs $\left(m,n\right)$ satisfy $\varphi \left(m,n\right)\le x$ with $\left(m,n\right)=1$ . Using analytic number theory methods such as the hyperbolic summation method and convolution method, we obtain an asymptotic expression for the distribution of $\varphi \left(m,n\right)$ .