Abstract
We obtain asymptotic formulas for the sums \( {\sum}_{n_1,\dots, {n}_k\leqslant x} \) f((n1, . . . , nk)) and \( {\sum}_{n_1,\dots, {n}_k\leqslant x} \) 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) = \( {\sum}_{p^{\nu}\Big\Vert {n}^{v^{\ell }}}\mathrm{investigated} \) by Duncan (1962) and Hassani (2018), and the functions A(n) = \( {\sum}_{p^{\nu}\Big\Vert n} vp, \) 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.
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László Tóth was financed by NKFIH in Hungary within the framework of the 2020-4.1.1-TKP2020 3rd thematic programme of the University of Pécs.
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Bordellès, O., Tóth, L. Additive arithmetic functions meet the inclusion–exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers. Lith Math J 62, 150–169 (2022). https://doi.org/10.1007/s10986-022-09565-w
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DOI: https://doi.org/10.1007/s10986-022-09565-w