Sums of digits and almost primes
type: Item
instance of: scholarly article
title: Sums of digits and almost primes
zbMATH Open document ID: 0858.11050
DOI: 10.1007/BF01444238
author: Christian Mauduit
author: Étienne Fouvry
published in: Mathematische Annalen
publication date: 1996-11-26T00:00:00Z
full work available at URL: https://eudml.org/doc/183004
review text: This paper is devoted to the problem of existence of almost primes in sets of integers generated by finite automata (where words are identified with integers via their expansion in a base). It is proved that for every irreducible automaton there is a constant \(k\) such that the generated sequence contains infinitely many numbers having at most \(k\) prime factors. The sets defined by \(\{n:s(n) \equiv b \pmod q\}\), where \(s(n)\) is the sum of binary digits of \(n\), are particular cases, and these are investigated in detail. It is proved that these always contain infinitely many numbers (even \(\gg x/ \log x\) below \(x)\) with at most two prime factors. If \({\mathcal Q}\) is a set of residue classes modulo \(q\) and \(|{\mathcal Q} |\geq 0.722q\), then there are infinitely many primes \(p\) such that the residue of \(s(p)\) modulo \(q\) is in \({\mathcal Q}\). These results are achieved by proving (analytically) a statistical result on the joint distribution of \(n\) modulo \(d\) and \(s(n)\) modulo \(q\), analogous to the Bombieri-Vinogradov theorem on primes, and applying weighted sieve methods.
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 931974
zbMATH Keywords: sets of integers generated by finite automata
zbMATH Keywords: digital sums
zbMATH Keywords: weighted sieve methods
zbMATH Keywords: almost primes
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 931974