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

Abstract

AbstractLet $$k\ge 2$$ k ≥ 2 be a fixed integer. We consider sums of type $$\sum _{n_1\cdots n_k\le x} F(n_1,\ldots ,n_k)$$ ∑ n 1 ⋯ n k ≤ x F ( n 1 , … , n k ) , taken over the hyperbolic region $$\{(n_1,\ldots ,n_k)\in {\mathbb {N}}^k: n_1\cdots n_k\le x\}$$ { ( n 1 , … , n k ) ∈ N k : n 1 ⋯ n k ≤ x } , where $$F:{\mathbb {N}}^k\rightarrow {\mathbb {C}}$$ F : N k → C is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations $$\sum _{n_1\cdots n_k\le x} f((n_1,\ldots ,n_k))$$ ∑ n 1 ⋯ n k ≤ x f ( ( n 1 , … , n k ) ) and $$\sum _{n_1\cdots n_k\le x} f([n_1,\ldots ,n_k])$$ ∑ n 1 ⋯ n k ≤ x f ( [ n 1 , … , n k ] ) , involving the GCD and LCM of the integers $$n_1,\ldots ,n_k$$ n 1 , … , n k , where $$f:{\mathbb {N}}\rightarrow {\mathbb {C}}$$ f : N → 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 $$k=2$$ k = 2 .