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

TL;DR

This paper derives an asymptotic formula for the mean value of a specific multiplicative function involving divisors and prime factors, using elementary methods to analyze its behavior.

Contribution

It provides a new asymptotic formula for the mean value of the function D_{k,ω}(n), combining divisor counts and prime divisor counts with elementary techniques.

Findings

Derived an explicit asymptotic formula for the mean value of D_{k,ω}(n)

Demonstrated the effectiveness of elementary methods in multiplicative number theory

Enhanced understanding of the distribution of divisor functions relative to prime factors

Abstract

For a fixed integer $k$, we define the multiplicative function \[D_{k,\omega}(n) := \frac{d(n)}{k^{\omega(n)}}, \]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.