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 $M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k))$ for any large real number $x\geq 5$, where $f$ is any arithmetical function. Let $ϕ$, and $ψ$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {\rm id})$, $M_r(x;ϕ)$ and $M_r(x;ψ)$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{\rm id})$ for any large positive number $x>5$ satisfying $x=[x]+\frac{1}{2}$.

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