On sums of logarithmic averages of gcd-sum functions

TL;DR

This paper derives asymptotic formulas for weighted averages of gcd-sum functions involving logarithms, focusing on multiplicative functions and Dirichlet series related to Anderson--Apostol sums.

Contribution

It provides new asymptotic formulas for gcd-sum functions with logarithmic weights for various multiplicative functions and explores related Dirichlet series.

Findings

Asymptotic formulas for sums involving gcd and logarithmic weights.

Results for specific multiplicative functions like id, φ, and their variants.

Formulas for Dirichlet series with Anderson--Apostol sum coefficients.

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 $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $\phi$, $id_{1+a}$ and $\phi_{1+a}$ 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.