Short interval results for a class of integers
type: Item
instance of: scholarly article
title: Short interval results for a class of integers
zbMATH Open document ID: 1099.11049
published in: Monatshefte für Mathematik
publication date: 2004-07-07T00:00:00Z
review text: Integers whose prime factorizations are subject to various constraints have long been a subject of fascination for number theorists. In particular, a great deal of effort has been dedicated to understanding the distribution of such integers. In this paper, the author studies the distribution of ``\(\mathcal G\)-numbers'' in short intervals. Suppose that \(\mathcal G\) is a set of positive integers which has a finite subset \(\{ g_1, \dots, g_s \}\) with \(\gcd(g_1, \dots, g_s) = 1\). We call a positive integer \(n\) a \textit{\(\mathcal G\)-number}, if all the exponents \(a_1, \dots, a_s\) in its prime factorization \(n = p_1^{a_1}p_2^{a_2} \cdots p_s^{a_s}\) belong to \(\mathcal G\). For example, when \(\mathcal G = \{1, 2, \dots, k - 1\}\), the \(\mathcal G\)-numbers are the \(k\)-free integers; and when \(\mathcal G = \{k, k + 1, \dots \}\), the \(\mathcal G\)-numbers are the \(k\)-full integers. With \(\mathcal G\) as above, we write \(g = \min \mathcal G\) and \(\delta(n; \mathcal G)\) for the characteristic function of the \(\mathcal G\)-numbers. The paper studies the range of \(y\) for which one has an asymptotic formula of the form \[ \sum_{x < n \leq x + y} \delta(n; \mathcal G) = yx^{-\alpha}(c + o(1)) \qquad (x \to \infty); \tag{1} \] here \(\alpha = 1 - g^{-1}\) and \(c = c(\mathcal G)\) is a constant depending only on the set \(\mathcal G\). It is easy to show that one can take \(y = x^{\theta}\) for any fixed \(\theta > \alpha + (g + 1)^{-1}\), and the primary goal is to relax the requirement for \(\theta\). The author shows that such results on the distribution of \(\mathcal G\)-numbers can be derived from results on two classical lattice point problems: the problem of estimating the number of lattice points close to a smooth plane curve; and the problem of estimating the remainders in asymptotic formulas of the form \[ \mathop{\sum \; 1}_{n_1^{u_1} \cdots n_r^{u_r} \leq x} = H(x; u_1, \dots, u_r) + O \left( x^{\beta} \right), \tag{2} \] where \(H(x; …
reviewed by: Angel V. Kumchev
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 2078307
zbMATH Keywords: G-numbers
zbMATH Keywords: powerfull numbers
zbMATH Keywords: powerfree numbers
zbMATH Keywords: distribution of lattice points
zbMATH Keywords: multidimensional divisor problems
zbMATH Keywords: short intervals
author: Wenguang Zhai
MaRDI profile type: Publication
full work available at URL: s00605-003-0006-9
OpenAlex ID: W2083232895
DOI: 10.1007/S00605-003-0006-9
description: scientific article; zbMATH DE number 2078307