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

Abstract

Let $k\ge 2$ be a fixed integer. We consider sums of type ${\sum }_{n_1^2+\cdots +n_k^2\le x}F(n_1,\dots ,n_k)$, taken over the $k$-dimensional spherical region $\{(n_1,\dots ,n_k)\in {\mathbb{Z}}^k:n_1^2+\cdots +n_k^2\le x\}$, where $F:{\mathbb{Z}}^k\to \mathbb{C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations ${\sum }_{n_1^2+\cdots +n_k^2\le x}f((n_1,\dots ,n_k))$ and ${\sum }_{n_1^2+\cdots +n_k^2\le x}f([n_1,\dots ,n_k])$, involving the GCD and LCM of the integers $n_1,\dots ,n_k$, where $f:\mathbb{N}\to \mathbb{C}$ belongs to certain classes of functions.

22 pages, revised

References in corpus (3)

• Hyperbolic summation for functions of the GCD and LCM of several integers
• On certain sums of arithmetic functions involving the gcd and lcm of two positive integers
• Additive arithmetic functions meet the inclusion-exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers