On the distribution of powerful numbers
type: Item
instance of: scholarly article
title: On the distribution of powerful numbers
zbMATH Open document ID: 0893.11036
DOI: 10.1007/BF02940832
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1998-08-10T00:00:00Z
review text: Let (1) \(n = p_1^{\alpha_1}\ldots p_r^{\alpha_r} (p_1 < \ldots < p_r)\) be the canonical decomposition of \(n\). If \(k,\ell \geq 2\) are given integers, then \(n\) is \(k\)-full (or powerful of type \(k\)) if \(n=1\) or if in (1) \(\alpha_j \geq k (j = 1,\ldots,r)\), and \(n\) is \(\ell\)-free if \(n=1\) or if in (1) \(\alpha_j \leq \ell-1 (j = 1,\ldots,r)\). Let \(\Delta_k(x)\) denote the error term in the asymptotic formula for the number of \(k\)-full integers not exceeding \(x\), and let \(N_{k,\ell}(x) (1 < k < \ell)\) denote the number of integers not exceeding \(x\) which are simultaneously \(k\)-full and \(\ell\)-free. The author is interested in upper bounds for \(\Delta_k(x)\) and \(\Delta_{k,\ell}(x,h)\), which represents the error term in the expression for \(N_{k,\ell}(x+h) - N_{k,\ell}(x)\) with suitable \(h>0\). The function \(\Delta_k(x)\) has been extensively studied in the literature, and some of the results are presented in the introductory part of the paper. The author proves five theorems, based on ten lemmas involving estimates in the general divisor problem (the generating function being \(\zeta(a_1s)\ldots\zeta(a_rs)\) for given integers \(1\leq a_1\leq\ldots\leq a_r\)). Four of the theorems involve upper bounds for \(\Delta_{k,\ell}(x,h)\) when \(k = 2,3,4,5\). On the other hand Theorem 1 gives the bounds \[ \Delta_5(x) \ll x^{54/605}\log^8x,\quad \Delta_6(x) \ll x^{67/843}\log^9x, \quad \Delta_7(x) \ll x^{1/14}\log^{10}x. \] Previously \textit{E. Krätzel} [Math. Nachr. 114, 97-104 (1983; Zbl 0545.10030)] obtained \(\Delta_5(x) \ll x^{6/65+\varepsilon}\) and \(\Delta_6(x) \ll x^{13/162+\varepsilon}\), while \textit{P. Shiu} and \textit{A. Ivić} [Ill. J. Math. 26, 576-590 (1982; Zbl 0484.10024)] proved that \(\Delta_7(x) \ll x^{1/14+\varepsilon}\).
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N45
zbMATH DE Number: 1114175
zbMATH Keywords: powerful numbers
zbMATH Keywords: \(\ell\)-free numbers
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: divisor problem
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: \(k\)-full numbers
author: Hartmut Menzer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1114175