On the number of prime factors of integers without large prime divisors
type: Item
instance of: scholarly article
title: On the number of prime factors of integers without large prime divisors
zbMATH Open document ID: 0621.10028
DOI: 10.1016/0022-314X(87)90017-5
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Omega\) (n) denote the total number of prime factors of n. The Erdős-Kac theorem asserts that the distribution of \(\Omega\) (n) on the set \(S(x)=\{n\in {\mathbb{N}}:\) \(n\leq x\}\) is essentially Gaussian with mean log log x and standard division \(\sqrt{\log \log x}\). The purpose of this paper is to prove an analogue of this theorem with S(x) replaced by \(S(x,y)=\{n\in {\mathbb{N}}:\) \(n\leq x\), p \(| n\Rightarrow p\leq y\}\). the author's result is an estimate with an error term which is uniform in the range log \(x\geq (\log y)^{21}\).
reviewed by: W. E. Briggs
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4006353
zbMATH Keywords: number of prime factors
zbMATH Keywords: integers without large prime divisors
zbMATH Keywords: total number of prime factors
zbMATH Keywords: Erdős-Kac theorem
author: Adolf Hildebrand
MaRDI profile type: Publication
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full work available at URL: 0022-314x(87)90017-5
OpenAlex ID: W2083780002
description: scientific article; zbMATH DE number 4006353