Exponential divisor problem and exponentially squarefree integers
type: Item
instance of: scholarly article
title: Exponential divisor problem and exponentially squarefree integers
zbMATH Open document ID: 0855.11047
author: Jie Wu
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/231255
full work available at URL: jtnb7-1.html
review text: An exponential divisor of an integer \(n\geq 2\) with prime factorization \(n= p_1^{\mu_1} \dots p_k^{\mu_k}\) is a divisor of the form \(d= p_1^{\nu_1} \dots p_k^{\nu_k}\) where, for each \(i\), \(\nu_i\) is a divisor of \(\mu_i\); \(n\) is called exponentially squarefree if each \(\mu_i\) is squarefree. Let \(\tau^{(e)} (n)\) be the number of exponential divisors of \(n\), and let \(q^{(e)} (n)\) denote the characteristic function of the exponentially squarefree integers, with the convention \(\tau^{(e)} (1)= q^{(e)} (1) =1\). Improving results of \textit{M. V. Subbarao} [in: Theory of arithmetic functions, Proc. Conf. Western Michigan Univ. 1971, Lect. Notes Math. 251, 247-271 (1972; Zbl 0237.10009)], the author establishes the estimates \[ \sum_{n\leq x} \tau^{(e)} (n)= A_1 x+ A_2 x^{1/2}+ O(x^{2/9} \log x) \] and \[ \sum_{n\leq x} q^{(e)} (n)= Bx+ O(x^{1/4} \exp \{-C(\log x)^{3/5} (\log \log x)^{-1/5} \}), \] where \(A_1\), \(A_2\), \(B\), and \(C\) are suitable constants. The method of proof is different from, and simpler than, that of Subbarao. For example, to establish the first estimate, the author represents the function \(\tau^{(e)}\) as a convolution \(\tau_{1,2} *f\), where \(\tau_{1,2}\) is defined by \(\sum_{n\geq 1} \tau_{1,2} (n) n^{-s}= \zeta (s) \zeta (2s)\) and \(f\) is asymptotically small in the sense that the Dirichlet series \(\sum_{n\geq 1} f(n) n^{-s}\) is absolutely convergent in the half-plane \(\sigma> 1/5\). The desired estimate then follows from known estimates for the sums \(\sum_{n\leq x} \tau_{1,2} (n)\).
reviewed by: Adolf Hildebrand
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 858020
zbMATH Keywords: generalized divisor function
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: exponential divisor
zbMATH Keywords: exponentially squarefree integers
MaRDI profile type: Publication
OpenAlex ID: W2317184996
cites work: Sur certaines fonctions additives à valeurs entières
cites work: On the difference between consecutive squarefree integers
cites work: Q3220660
cites work: On the Difference between Consecutive Squarefree Numbers
cites work: Q5646958
DOI: 10.5802/JTNB.136
description: scientific article; zbMATH DE number 858020