Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM

TL;DR

This paper derives asymptotic formulas with remainder estimates for sums over k-dimensional spheres involving arithmetic functions of the GCD and LCM of integer vectors, extending understanding of their distribution.

Contribution

It provides new asymptotic formulas with explicit error terms for spherical sums involving GCD and LCM functions, generalizing previous results to higher dimensions.

Findings

Asymptotic formulas for sums involving GCD and LCM over spherical regions.

Explicit remainder terms in the asymptotic estimates.

Extension of classical number theory sums to higher dimensions.

Abstract

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.