On sums of gcd-sum functions

Let gcd(j, k) denote the greatest common divisor of integers k and j. Define A(f)(x) = Sigma(k <= x) 1/k Sigma(k)(j=1) f (gcd (j, k)), for any arithmetical function f. We shall derive several asymptotic expansions of the error terms E-f(x) for the partial sums of A(f)(k) with f= id, id(1+a) (-1 < a < 0), phi, and psi, where phi and psi, are the Euler totient function and the Dedekind function, respectively. Furthermore, we establish several formulas concerning the sum function A(f)(x), and deduce several related results.