Sums of Logarithmic Averages of gcd-Sum Functions

Abstract

Let $\gcd (k,j)$ be the greatest common divisor of the integers k and j. For any arithmetic function f, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is

$$\begin{aligned} \sum _{k\le x}\frac{1}{k} \sum _{j=1}^{k}f(\gcd (k,j)) \log j. \end{aligned}$$

More precisely, we give asymptotic formulas for various multiplicative functions such as $f=\mathrm{id}$, $\phi$, $\mathrm{id}_{1+a}$ and $\phi _{1+a}$ with $-1<a<0$. We also consider several formulas of Dirichlet series associated with Anderson–Apostol sums.