Sums of averages of gcd-sum functions

Let gcd⁡(k,j) be the greatest common divisor of the integers k and j. We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is ∑k≤x1kr+1∑j=1kjrf(gcd⁡(k,j)) with f=id,ϕ,ϕs,ψ and ψs for any fixed positive integers r and s, where ϕ,ϕs,ψ and ψs are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the error term of the gcd-sum functions ∑k≤x1kr+1⋅∑j=1kjrϕ(gcd⁡(k,j)) and ∑k≤x1kr+1∑j=1kjrψ(gcd⁡(k,j)).