On the distribution of elements of semigroups of natural numbers
type: Item
instance of: scholarly article
zbMATH Open document ID: 1306.11014
author: Yu. N. Shteĭnikov
publication date: 2013-11-19T00:00:00Z
full work available at URL: cheb51
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 6229108
zbMATH Keywords: multiplicative subgroups
zbMATH Keywords: Buchstab function
zbMATH Keywords: smooth numbers
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: distribution of integers without large prime factors
zbMATH Keywords: short intervals
MaRDI profile type: Publication
title: On the distribution of elements of semigroups of natural numbers
published in: Chebyshevskiĭ Sbornik
review text: Let \(A\) be a multiplicative subgroup of positive integers such that \(|\{n\in A; n\leq q\}|<g^\nu\) for some real \(q\) and \(\nu<1\). The author proves the upper estimates for \(f(x)=|A\cap [1,x]|\) of the following type for \(x=(\log q)^u\):NEWLINENEWLINE (1) if \(\log\log x=o(\log\log q)\) then NEWLINE\[NEWLINE\frac{f(x)}{x}\leq \exp\{-(C+o(1))u(1-\nu^2)\log(u(1-v)^2)\}NEWLINE\]NEWLINE with an absolute constant \(C\);NEWLINENEWLINE (2) if \(\gamma=(\log\log x)/(\log\log q)\) and \(\log x=o(\log q)\) then NEWLINE\[NEWLINEf(x)\leq x^{1-\max\{L_\gamma,C_\gamma\}+o(1)}\quad\text{as}\;q\to\infty,NEWLINE\]NEWLINE with explicitly given constants \(L_\gamma\) and \(C_\gamma\) depending on \(\gamma\) and \(\nu\).
reviewed by: Štefan Porubský
description: scientific article; zbMATH DE number 6229108