Additive arithmetic functions meet the inclusion–exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers

Abstract

AbstractWe obtain asymptotic formulas for the sums $$ {\sum}_{n_1,\dots, {n}_k\leqslant x} $$ ∑ n 1 , … , n k ⩽ x f((n1, . . . , nk)) and $$ {\sum}_{n_1,\dots, {n}_k\leqslant x} $$ ∑ n 1 , … , n k ⩽ x f([n1, . . . , nk]), involving the GCD and LCM of the integers n1, . . . , nk, where f belongs to certain classes of additive arithmetic functions. In particular, we consider the generalized omega function Ωℓ(n) = $$ {\sum}_{p^{\nu}\Big\Vert {n}^{v^{\ell }}}\mathrm{investigated} $$ ∑ p ν ‖ n v ℓ investigated by Duncan (1962) and Hassani (2018), and the functions A(n) = $$ {\sum}_{p^{\nu}\Big\Vert n} vp, $$ ∑ p ν ‖ n vp , A∗(n) = ∑p ∣ np, B(n) = A(n) − A∗(n) studied by Alladi and Erdős (1977). As a key auxiliary result, we use an inclusion–exclusion-type identity.