Almost primes whose expansion in base \(r\) misses some digits
type: Item
instance of: scholarly article
title: Almost primes whose expansion in base \(r\) misses some digits
zbMATH Open document ID: 0957.11039
author: Christian Mauduit
author: Cécile Dartyge
published in: Journal of Number Theory
publication date: 2001-03-26T00:00:00Z
review text: This paper is devoted to establishing the existence of almost primes, having at most \(k=k(r)\) prime factors, among the integers whose representation in base~\(r\) does not include certain specified digits. Some of their results apply to numbers whose digits to base~\(r\) are all either 0 or~1. By means of the linear sieve they obtain such a result with \(k(3)=4\), \(k(4)=5\), and where \(k(r) \sim 8r/\pi\) as \(r \to \infty\). If the number \(t\) of permitted digits is somewhat larger then better values for \(k(r)\) follow from using the sieve with weights as described by the reviewer in [Number Theory, Trace Formulas and Discrete Groups, Academic Press, Boston, 289-308 (1989; Zbl 0682.10038)]. If \(t > \sqrt r\) then \(k(r)=5\) is allowed, and if \(t \geq r^{0.68381}\) then \(k(r)=3\) is obtained. The results on the associated ``level of distribution'' needed for these applications of sieve methods are obtained by developing techniques used by \textit{E. Fouvry} and \textit{C. Mauduit} [Acta Arith. 77, 339-351 (1996; Zbl 0869.11073)] and by \textit{P. Erdős, C. Mauduit} and \textit{A. Sárközy} [Discrete Math. 200, 149-154 (1999; Zbl 0945.11006)].
reviewed by: George Greaves
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 1443342
zbMATH Keywords: almost primes
zbMATH Keywords: missing digits
zbMATH Keywords: sieve
MaRDI profile type: Publication
full work available at URL: jnth.1999.2458
OpenAlex ID: W2042003909
cites work: On arithmetic properties of integers with missing digits. I: Distribution in residue classes
cites work: On arithmetic properties of integers with missing digits. II: Prime factors
cites work: Squarefree values of polynomials all of whose coefficients are 0 and 1
cites work: Sums of digits and almost primes
cites work: Méthodes de crible et fonctions sommes des chiffres
cites work: Q3901584
cites work: Q4731255
cites work: An Application of a Theorem of Barban, Davenport and Halberstam
cites work: Q4052222
cites work: On the arithmetic structure of the integers whose sum of digits is fixed
cites work: Topics in multiplicative number theory
cites work: Q4332879
DOI: 10.1006/JNTH.1999.2458
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description: scientific article; zbMATH DE number 1443342