On sums of weighted averages of $\gcd$-sum functions

Abstract

Let $\gcd (j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd (j,k))$ and the function ${\sum }_{d|k,d^s|j}(f\ast \mu )(d)$ for any positive integers $j$ and $k$, namely $$\sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^k{j}^{r}f(\gcd (j,k))\text{and}\sum _{k\le x}\frac{1}{k^{s(r+1)}}\sum _{j=1}^{k^s}{j}^r\sum _{\begin{array}{c}d|k{d}^s|j\end{array}}(f\ast \mu )(d),$$ with any fixed integer $s>1$ and any arithmetical function $f$. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of $\gcd$-sum functions $f(\gcd (j,k)).$

30 pages