Communications of the
Korean Mathematical Society CKMS

In this paper, we investigate the sums of the elements in the finite set $\{x^{k}:1\leq x\leq\frac{n}{m},\gcd_u(x,n)=1\}$, where $k$, $m$ and $n$ are positive integers and $\gcd_u(x,n)$ is the unitary greatest common divisor of $x$ and $n$. Moreover, for some cases of $k$ and $m$, we can give the explicit formulae for the sums involving some well-known arithmetic functions.

Keywords: Arithmetic function, unitary divisor, M\"{o}bius function, M\"{o}bius inversion formula

MSC numbers: Primary 11A25