On 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 arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\sum _{k\le x}\frac{1}{k}\sum _{j=1}^{k}f(\gcd (k,j))\log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $\varphi$, $i{d}_{1+a}$ and ${\varphi }_{1+a}$ with $-1<a<0$. We also establish some formulas of Dirichlet series having coefficients of the sum function ${\sum }_{j=1}^k{s}_k(j)\log j$ where $s_k(j)$ is Anderson--Apostol sums.

21 Pages