Sums of weighted averages of gcd-sum functions

Abstract

Let [Formula: see text] be the greatest common divisor of the integers [Formula: see text] and [Formula: see text]. In this paper, we give several interesting asymptotic formulas for weighted averages of the [Formula: see text]-sum function [Formula: see text] and the function [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text], namely [Formula: see text] with any fixed integer [Formula: see text] and any arithmetical function [Formula: see text]. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of [Formula: see text]-sum functions [Formula: see text]