On the asymptotic behavior of large prime factors of integers
type: Item
instance of: scholarly article
title: On the asymptotic behavior of large prime factors of integers
zbMATH Open document ID: 0419.10042
author: Krishnaswami Alladi
author: P. Erdős
published in: Pacific Journal of Mathematics
publication date: 1979-01-01T00:00:00Z
review text: The paper under review continues the work of Hardy and Ramanujan on round numbers and related topics. The authors study the functions \(p_1(n)\), \(p_2(n)\dots\) defined as the biggest prime factor of n, and the next smaller prime factor and so on. More precisely let \(n=\prod_{i=1}^rp_i^{\alpha_i}\) where \(p_1> p_2> \dots\) and \(\alpha_i> 0\). \(A(n)=\Sigma\alpha_1\), \(A^*(n)=\Sigma p_i\), \(Omega(n)=\Sigma\alpha_1\), \(\omega(n)=r\), \(P_1(n)=P_1^*(n)=p_1\), \(P_k^*(n)=p_k\) for \(k\leq\omega(n)\) and zero for \(k> \omega(n)\), \(P_k(n)=P_1\left(\frac n{P_1(n)P_2(n)\dots P_{k-1}(n)}\right)\) for \(1< k\leq\Omega(n)\) and zero for \(k> \Omega(n)\). The functions \(\Omega(n)\) and \(\omega(n)\) were studied by Hardy and Ramanujan who proved that their normal order is \(\log\log n\). The authors make a comparative study of A(n), \(A^*(n)\), \(P_k(n)\) and \(P_k^*(n)\). With this in view they introduce \[ S_1(x,k) = \sum_{2\leq n\leq x} \frac{A(n)-P_1(n)- \dots -P_{k-1}(n)}{P_1(n)},\qquad S_2(x,k) = \sum_{2\leq n\leq x}\frac{A^*(n)-P_1^*(n)-\dots-P_{k-1}^*(n)}{P_1(n)}, \] \[ S_3(x,k) = \sum_{2\leq n\leq x} \frac{P_k(n)}{P_1(n)},\text{ and } S_4(k,x) = \sum_{2\leq n\leq x} \frac{P_k^*(n)}{P_1(n)}, \] where \(k\geq1\) is any fixed positive integer. In anearlier paper they proved that as \(x\to\infty\), \[ \sum_{1\leq n\leq x}(A(n)-P_1(n)-\dots-P_{k-1}(n))\sim \sum_{1\leq n\leq x}P_k(n)\sim\sum_{1\leq n\leq x}\P_k^*(n)\sim\sum_{1\leq n\leq x}(A^*(n)-P_1^*(n)-\dots-\P_{k-1}^*(n))\sim a_k(\log x)^{-k}x^{1+1/k}, \] where \(a_k\) is a positive constant depending on k. They also proved \[ \sum_{1\leq n\leq x}(A(n)-A^*(n))=x\log\log x+0(x). \] To these they add another set of interesting results. Namely that as \(x\to\infty\), \[ S_1(x,k)\sim S_2(x,k)\sim S_3(x,k)\sim S_4(x,k)\sim a_k'(\log x)^{1-k}x, \] where \(a_k'\) is a positive constant depending only on k. These results give satisfactory information of the asymptotic behaviour (i.e. average order, normal order etc.) o…
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 3652457
zbMATH Keywords: Hardy-Ramanujan theorem
zbMATH Keywords: round numbers
zbMATH Keywords: large prime factors
zbMATH Keywords: average order
zbMATH Keywords: normal order
MaRDI profile type: Publication
full work available at URL: pjm.1979.82.295
OpenAlex ID: W2106281666
Wikidata QID: Q105659348
DOI: 10.2140/PJM.1979.82.295
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description: scientific article; zbMATH DE number 3652457