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

Abstract

Let kge 2 be a fixed integer. We consider sums of type sum _{n_1cdots n_kle x} F(n_1,ldots ,n_k), taken over the hyperbolic region {(n_1,ldots ,n_k)in {mathbb {N}}^k: n_1cdots n_kle x}, where F:{mathbb {N}}^krightarrow {mathbb {C}} is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations sum _{n_1cdots n_kle x} f((n_1,ldots ,n_k)) and sum _{n_1cdots n_kle x} f([n_1,ldots ,n_k]), involving the GCD and LCM of the integers n_1,ldots ,n_k, where f:{mathbb {N}}rightarrow {mathbb {C}} belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k=2.