Sums of averages of gcd-sum functions II

Abstract

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

14 pages