Sums of Averages of GCD-Sum Functions II

Let gcd(k,j) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define Mr(x;f):= Sigma k <= x1kr+1 Sigma j=1k jrf(gcd(j,k))for any large real number x >= 5, where f is any arithmetical function. Let phi, and psi denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of Mr(x;id), Mr(x;phi) and Mr(x;psi). Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of Mr(x;id) for any large positive number x>5 satisfying x=[x]+<mml:mfrac>12</mml:mfrac>.