Distribution of the number of prime factors with a given multiplicity

TL;DR

This paper analyzes the distribution of prime factors with specific multiplicities in integers, providing explicit formulas, generating functions, and asymptotic bounds, extending classical results on additive functions of prime factors.

Contribution

It derives the density distribution for the number of prime factors with fixed multiplicity, introduces a product-form generating function, and generalizes to all additive functions of this type.

Findings

Derived explicit densities $e_{k,m}$ for prime factors with multiplicity $k$.

Established the generating function as an entire function with a product representation.

Provided methods for numerical calculation and asymptotic bounds for $e_{k,m}$.

Abstract

Given an integer $k\ge2$, let $\omega_k(n)$ denote the number of primes that divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of those integers $n$ for which $\omega_k(n)=m$ for every integer $m\ge0$. We also show that the generating function $\sum_{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod_{p} \bigl(1+{(p-1)(z-1)}/{p^{k+1}} \bigr)$; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum_{j=2}^\infty a_j \omega_j(n)$; when $a_j=j-1$ this recovers a classical result of R\'enyi on the distribution of $\Omega(n)-\omega(n)$.