An asymptotic formula for the gcd-sum function of Jordan's totient function

Abstract:

Let $\gcd(k, j)$ denote the greatest common divisor of the positive integers $k$ and $j$, $r$ be any fixed positive integer. In this paper, by using the identity of Kiuchi on $M_r(x; f)$ together with analytic method, we present asymptotic formulas of $ M_r(x;J_k)$. This complements and strengthens the corresponding result obtained by Kiuchi and Saad eddin in 2018.