On the distribution of square-full and cube-full numbers
type: Item
instance of: scholarly article
title: On the distribution of square-full and cube-full numbers
zbMATH Open document ID: 0838.11059
DOI: 10.1007/BF01585911
published in: Monatshefte für Mathematik
publication date: 1996-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/178696
review text: An integer \(n\) is said to be \(k\)-full if \(p^k\) divides \(n\) for every prime divisor \(p\) of \(n\). Much has been written on the distribution of such numbers which was first studied by P. Erdös and G. Szekeres in 1935. In 1984 the reviewer [Glasg. Math. J. 25, 127-134 (1984; Zbl 0526.10035)] initiated the short interval problem, which can be transformed to the study of a certain exponential sum in an interval. The author [Lattice points (1988; Zbl 0675.10031)], \textit{D. R. Heath-Brown} [Math. Proc. Camb. Philos. Soc. 110, 1-3 (1991; Zbl 0735.11043)], \textit{H. Q. Liu} [Acta Arith. 64, 129-149 (1993; Zbl 0778.11050)] and the reviewer [Math. Proc. Camb. Philos. Soc. 112, 1-5 (1992; Zbl 0769.11032)] have made recent significant progress in this problem when \(k = 2,3\). An \(\ell\)-free number is one which is not divisible by \(p^\ell\) for any prime \(p\), and in this paper the author considers those \(k\)-full numbers which are also \(\ell\)-free, with \(\ell > k + 1\). He first uses Dirichlet series to derive asymptotic formulae for the number of such numbers up to \(x\). He then studies the short interval problem for \(k = 2,3\), giving significant results which are too complicated to be reported in full here.
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 819238
zbMATH Keywords: exponential sums
zbMATH Keywords: \(\ell\)-free \(k\)-full numbers
zbMATH Keywords: short interval problem
zbMATH Keywords: Dirichlet series
zbMATH Keywords: asymptotic formulae
author: Ekkehard Krätzel
reviewed by: Peter Shiu
MaRDI profile type: Publication
cites work: On a theorem of Erdős and Szekeres
cites work: Square-full numbers in short intervals
cites work: Q4175344
cites work: Estimates in the general divisor problem
cites work: On square-full numbers in short intervals
cites work: The number of squarefull numbers in an interval
cites work: Q4023789
cites work: Zur Anzahl quadratvoller Zahlen in kurzen Intervallen
cites work: Über die Anzahl quadratvoller Zahlen in kurzen Intervallen und ein verwandtes Gitterpunktproblem
cites work: On square-full integers in a short interval
cites work: Cube-full numbers in short intervals
cites work: Many-Dimensional Generalized Divisor Problems
cites work: Exponential Sums and the Riemann Zeta-Function II
Recommended article: On the distribution of square-full and cube-full integers
Recommended article: The distribution of cube-full numbers
Recommended article: On the distribution of square-full integers
Recommended article: Q3124122
Recommended article: The distribution of square-full integers
Recommended article: On the distribution of square-full numbers in arithmetic progressions
description: scientific article; zbMATH DE number 819238