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

TL;DR

This paper derives asymptotic formulas for sums involving additive arithmetic functions applied to the gcd and lcm of multiple integers, extending previous work with new classes of functions and an inclusion-exclusion approach.

Contribution

It introduces new asymptotic formulas for sums of additive functions over gcd and lcm, utilizing an inclusion-exclusion identity and analyzing functions like the generalized omega and related sums.

Findings

Derived asymptotic formulas for sums involving gcd and lcm.

Analyzed specific additive functions including the generalized omega function.

Utilized an inclusion-exclusion type identity for key auxiliary results.

Abstract

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 $f$ 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 $A(n)=\sum_{p^\nu \mid\mid n} \nu p$, $A^*(n)= \sum_{p \mid n} p$, $B(n)=A(n)-A^*(n)$ studied by Alladi and Erd\H{o}s (1977). As a key auxiliary result we use an inclusion-exclusion-type identity.