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

Abstract

We obtain asymptotic formulas for the sums {sum}_{n_1,dots, {n}_kleqslant x} f((n1, . , nk)) and {sum}_{n_1,dots, {n}_kleqslant 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}BigVert {n}^{v^{ell }}}mathrm{investigated} by Duncan (1962) and Hassani (2018), and the functions A(n) = {sum}_{p^{nu}BigVert 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.