论文标题
GCD和LCM的算术功能的$ k $二维球形总结的估计值
Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM
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_1^2+\ cdots+n_k^2 \ le x} f（n_1，\ ldots，n_k）$的总和，取代了$ k $ - diepermentional-demensional球形区域$ \ \ \ \ \ {（n_1，n_1，n_1，n_k \ ldots，n_k） n_k^2 \ le x \} $，其中$ f：{\ bbb z}^k \ to {\ bbb c} $是给定函数。特别是，我们推断出具有剩余术语的渐近公式的球形求和$ \ sum_ {n_1^2+ \ cdots+ n_k^2 \ le x} f（（（n_1，\ ldots，n_k））$和$ \ sum___________ { f（[n_1，\ ldots，n_k]）$，涉及整数的gcd和lcm $ n_1，\ ldots，n_k $，其中$ f：{\ bbb n} \ to {\ bbb c} $属于某些类别的功能类别。
Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the $k$-dimensional spherical region $\{(n_1,\ldots,n_k)\in {\Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x\}$, where $F:{\Bbb Z}^k\to {\Bbb C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations $\sum_{n_1^2+\cdots+ n_k^2\le x} f((n_1,\ldots,n_k))$ and $\sum_{n_1^2+\cdots+ n_k^2\le x} f([n_1,\ldots,n_k])$, 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.
