On a problem of Erdős and Alladi
type: Item
instance of: scholarly article
title: On a problem of Erdős and Alladi
zbMATH Open document ID: 0531.10046
DOI: 10.1007/BF01653241
author: Adolf Hildebrand
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178167
review text: Let p(n) denote the largest prime factor of n and set, for an arithmetic function f, \(\psi_ f(x,y)=\sum_{n\leq x,p(n)\leq y}f(n),\quad M_ f(x)=\sum_{n\leq x}f(n).\) It is shown that if f is completely multiplicative or ''squarefree multiplicative'' (i.e. multiplicative and supported by squarefree numbers) and satisfies \(| f| \leq 1\), then \[ | \psi_ f(x,y)| \ll \psi_ 1(x,y)\{\sup(| M_ f(t)| t^{-1}\quad:\quad \sqrt{y}\leq t\leq \min(x,y^ 2))+\log^{- 1}x\int^{y}_{1}| M_ f(t)| t^{-2}dt\} \] holds for all \(x\geq y\geq 2\), the implied constant being absolute. For the Möbius function \(\mu\), this implies \[ \sup_{y\geq 2}| \psi_{\mu}(x,y)| /\psi_ 1(x,y)\quad \ll \quad 1/\log \log x \] and thus proves a conjecture of Erdős.
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3841985
zbMATH Keywords: multiplicative function
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: Möbius function
zbMATH Keywords: largest prime factor
zbMATH Keywords: Dickman-de Bruijn function
MaRDI profile type: Publication
cites work: Number theory. Proceedings of the Third Matscience Conference held at Mysore, India, June 3-6, 1981
cites work: Asymptotic Estimates of Sums Involving the Moebius Function. II
cites work: Q5562701
description: scientific article; zbMATH DE number 3841985