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 ${\sum }_{n_1,\dots ,n_k\le x}f((n_1,\dots ,n_k))$ and ${\sum }_{n_1,\dots ,n_k\le x}f([n_1,\dots ,n_k])$ involving the gcd and lcm of the integers $n_1,\dots ,n_k$, where $f$ belongs to certain classes of additive arithmetic functions. In particular, we consider the generalized omega function ${\Omega }_{\ell }(n)={\sum }_{p^{\nu }\mid \mid n}{\nu }^{\ell }$ investigated by Duncan (1962) and Hassani (2018), and the functions $A(n)={\sum }_{p^{\nu }\mid \mid n}\nu p$, $A^{\ast }(n)={\sum }_{p\mid n}p$, $B(n)=A(n)-A^{\ast }(n)$ studied by Alladi and Erdős (1977). As a key auxiliary result we use an inclusion-exclusion-type identity.

22 pages