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We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x}f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le x} f([n_1,\ldots,n_k])$involving the gcd and lcm of the integers $n_1,\ldots,n_k$, where $f$ belongsto certain classes of additive arithmetic functions. In particular, we considerthe generalized omega function $\Omega_{\ell}(n)= \sum_{p^\nu \mid\mid n}\nu^{\ell}$ investigated by Duncan (1962) and Hassani (2018), and the functions$A(n)=\sum_{p^\nu \mid\mid n} \nu p$, $A^*(n)= \sum_{p \mid n} p$,$B(n)=A(n)-A^*(n)$ studied by Alladi and Erd\H{o}s (1977). As a key auxiliaryresult we use an inclusion-exclusion-type identity.

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