论文标题
GCD和LCM功能的双曲线总和几个整数
Hyperbolic summation for functions of the GCD and LCM of several integers
论文作者
论文摘要
令$ k \ ge 2 $为固定整数。我们考虑类型$ \ sum_ {n_1 \ cdots n_k \ le x} f（n_1，\ ldots，n_k）$的总和，取过了双曲线区域$ \ {（n_1，\ ldots，n_k，n_k）\ in {\ bb n} $ f：{\ bbb n}^k \ to {\ bbb c} $是给定函数。特别是，我们推断出具有剩余术语的渐近公式，以$ \ sum_ {n_1 \ cdots n_k \ le x} f（（（n_1，\ ldots，n_k））$和$ \ sum_ sum_1整数的lcm $ n_1，\ ldots，n_k $，其中$ f：{\ bbb n} \ to {\ bbb c} $属于某些类别的函数。我们的一些结果概括了作者以$ k = 2 $获得的结果。
Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1\cdots n_k\le x} F(n_1,\ldots,n_k)$, taken over the hyperbolic region $\{(n_1,\ldots,n_k)\in {\Bbb N}^k: n_1\cdots n_k\le x\}$, where $F:{\Bbb N}^k\to {\Bbb C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{n_1\cdots n_k\le x} f((n_1,\ldots,n_k))$ and $\sum_{n_1\cdots n_k\le x} f([n_1,\ldots,n_k])$, involving the GCD and LCM of the integers $n_1,\ldots,n_k$, where $f:{\Bbb N}\to {\Bbb C}$ belongs to certain classes of functions. Some of our results generalize those obtained by the authors for $k=2$.
