Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Distribution of the number of prime factors with a given multiplicity (2406.04574v1)

Published 7 Jun 2024 in math.NT

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 $\sum_{m=0}\infty e_{k,m}zm$ 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 $\Omega(n)-\omega(n)$.

Summary

We haven't generated a summary for this paper yet.