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 ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ⁡ ( k , j ) ) with f = id , ϕ , ϕ s , ψ and ψ s for any fixed positive integers r and s , where ϕ , ϕ s , ψ and ψ s 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 ≤ x 1 k r + 1 ⋅ ∑ j = 1 k j r ϕ ( gcd ⁡ ( k , j ) ) and ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r ψ ( gcd ⁡ ( k , j ) ) . [ABSTRACT FROM AUTHOR]