Asymptotic formula for the multiplicative function $\frac{d(n)}{k^{\omega (n)}}$

Abstract

For a fixed integer $k$, we define the multiplicative function \[D_{k,ω}(n) := \frac{d(n)}{k^{ω(n)}}, \]where $d(n)$ is the divisor function and $\omega (n)$ is the number of distinct prime divisors of $n$. The main purpose of this paper is the study of the mean value of the function $D_{k,\omega }(n)$ by using elementary methods.