Abstract
Let [Formula: see text] denote the greatest common divisor of integers [Formula: see text] and [Formula: see text]. Define [Formula: see text] for any arithmetical function [Formula: see text]. We shall derive several asymptotic expansions of the error terms [Formula: see text] for the partial sums of [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where [Formula: see text] and [Formula: see text] are the Euler totient function and the Dedekind function, respectively. Furthermore, we establish several formulas concerning the sum function [Formula: see text] and deduce several related results.
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