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

TL;DR

This paper derives asymptotic formulas and mean value results for weighted averages of gcd-sum functions and related sums involving arithmetical functions, extending understanding of their average behavior.

Contribution

It provides new asymptotic formulas and mean value results for weighted averages of gcd-sum functions and related sums for arbitrary arithmetical functions.

Findings

Asymptotic formulas for weighted averages of gcd-sum functions.

Mean value formulas for error terms in partial sums of gcd-sum functions.

Results hold for any fixed integer s > 1 and arbitrary arithmetical 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*\mu)(d) $ for any positive integers $j$ and $k$, namely $$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad \text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r} \sum_{\substack{d|k d^{s}|j}}(f*\mu)(d), $$ 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)). $