Hyperbolic summation for functions of the GCD and LCM of several integers

Abstract

Let $k\ge 2$ be a fixed integer. We consider sums of type ${\sum }_{n_1\cdots n_k\le x}F(n_1,\dots ,n_k)$, taken over the hyperbolic region $\{(n_1,\dots ,n_k)\in {\mathbb{N}}^k:n_1\cdots n_k\le x\}$, where $F:{\mathbb{N}}^k\to \mathbb{C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations ${\sum }_{n_1\cdots n_k\le x}f((n_1,\dots ,n_k))$ and ${\sum }_{n_1\cdots n_k\le x}f([n_1,\dots ,n_k])$, 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. Some of our results generalize those obtained by the authors for $k=2$.

15 pages, revised

References in corpus (3)

• On certain sums of arithmetic functions involving the gcd and lcm of two positive integers
• Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions
• Additive arithmetic functions meet the inclusion-exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers

Cited by in corpus (1)

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