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

TL;DR

This paper develops asymptotic formulas for sums over hyperbolic regions involving functions of the GCD and LCM of multiple integers, extending previous results for the case of two integers to higher dimensions.

Contribution

It introduces new asymptotic formulas with remainder terms for hyperbolic sums involving GCD and LCM of several integers, generalizing earlier two-variable results.

Findings

Derived asymptotic formulas for sums involving GCD and LCM

Extended previous two-variable results to higher dimensions

Provided remainder estimates for the asymptotic formulas

Abstract

Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1\cdots n_k\le x} F(n_1,\ldots,n_k)$, 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 $k=2$.