与酉最小公倍数有关的和函数

The Sum Function Related to the Least Unitary Common Multiple

摘要: 最大公因数和最小公倍数问题是数论的经典问题之一，那么对任意的正整数n，若$d|n$ 且$\left(d,\frac{n}{d}\right)=1$ ，我们就称d是n的酉除数(unitary divisor)，用${\left[m,n\right]}^{\ast }$ 表示两个数m, n的酉最小公倍数，本文主要考虑当${\left[m,n\right]}^{\ast }\le x$ 时，满足该条件的点$\left(m,n\right)$ 的和函数的渐近公式。

Abstract: The greatest common divisor and the least common multiple are one of the classical problems in number theory. For any positive integer n, if $d|n$ and $\left(d,\frac{n}{d}\right)=1$ , we call d is the unitary divisor of n. In this paper, using ${\left[m,n\right]}^{\ast }$ to represent the unitary least common multiple of two numbers m, n, and this article mainly considers the asymptotic formula of the sum function of points that satisfy this condition when ${\left[m,n\right]}^{\ast }\le x$ .