Sums of Averages of GCD-Sum Functions II

Abstract Let $$ \gcd (k,j) gcd ( k , j ) denote the greatest common divisor of integers k and j , let r be any fixed positive integer. Define $$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$ M r x ; f : = ∑ ≤ 1 + for large real number $$x\ge 5$$ ≥ 5 where f is arithmetical function. $$\phi ϕ $$\psi ψ Euler totient Dedekind function, respectively. In this paper, we refine asymptotic expansions $$M_r(x; \mathrm{id})$$ id $$M_r(x;{\phi })$$ $$M_r(x;{\psi . Furthermore, under Riemann Hypothesis simplicity zeros zeta-function, establish formula $$M_r(x;\mathrm{id})$$ $$x>5$$ > satisfying $$x=[x]+\frac{1}{2}$$ [ ] 2