关于Jordan函数的gcd和函数的渐近估计

将整数$k$ 和 $j$的最大公约数记为$gcd(k, j)$.设$k$为正整数, $f$为任意的算术函数, $r$是任一固定的整数.其中$n$为任意正整数. 对实数$x ge 2$, 我们定义与$f$相关联的gcd-和函数$M_r(x; f)$如下：$$M_r(x; f):=sumlimits_{k le x}frac{1}{k^{r+1}}sumlimits_{j=1}^k j^rf(gcd(k,j)).$$本论文中, 我们主要利用Kiuchi在2017年所得到的关于$M_r(x; f)$的一个恒等式, 以及初等和解析方法, 给出了$ M_r(x;J_k)$的渐近公式.若当函数$J_k$定义为$J_k(n):=n^kprodlimits_{p|n}(1-frac{1}{p^k})$,这加强了Kiuchi和Saad eddin在2018年所得到的结果

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

Let $gcd(k, j)$ denote the greatest common divisorof the positive integers $k$ and $j$, and let $r$ be any fixed positive integer.Let $M_r(x;f):=sum_{k le x}frac{1}{k^{r+1}}sum_{j=1}^k j^rf(gcd(k,j))$,where $xge 2$ is any large real number and $f$ is any arithmetical function.Let $J_k$ denote Jordan''s totient function that is defined for any integer$nge 1$ by $J_k(n):=n^kprodlimits_{p|n}(1-frac{1}{p^k})$.In this paper, by using the identity of Kiuchi on $M_r(x; f)$ together withanalytic method, we present asymptotic formulas of $ M_r(x;J_k)$.This complements and strengthens the corresponding result obtainedby Kiuchi and Saad eddin in 2018.