Some problems of analytic number theory on arithmetic semigroups
type: Item
instance of: scholarly article
title: Some problems of analytic number theory on arithmetic semigroups
zbMATH Open document ID: 1231.11114
author: Glyn Harman
author: Kaisa Matomäki
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1229624649
review text: Let \(E\) be a set of primes satisfying \[ (*)\quad \sum_{ p \in E, p \leq x} \frac{\log p}{p}=\tau \log x +O(1), \] for some positive constant \(\tau\). Let \(A\) be the multiplicative semigroup generated by the primes from \(E\). The authors study a variety of natural questions on the distribution of elements in \(A\), or multiplicative properties of elements of shifted sets \(A+k\). Questions on the counting function of \(A\) go back, for example, to \textit{E. Wirsing} [Arch. Math. 7, 263--272 (1956; Zbl 0075.25001)]. Questions on prime factors of shifted members of \(A\) have been raised for example in [the reviewer, Funct. Approximatio, Comment. Math. 35, 61--77 (2006; Zbl 1196.11139) and Bull. Lond. Math. Soc. 40, No. 1, 97--107 (2008; Zbl 1203.11064)]. Here we state some of the new results. A results for short intervals: Let \(1 \geq \tau > \frac{1}{2}\) and suppose that \((*)\) holds: then there exists \(\delta=\delta(\tau)>0\) and \(C(E)>0\) such that \[ \left|\{n\in A, x \leq n < x^{1-\delta}\}\right|>\frac{C(E)x^{1-\delta}}{(\log x)^{1-\tau}}. \] A result for shifted primes: Let \(k\neq 0\) be an even integer. There exist \(\tau_0<1\) such that if \(\tau_0 \leq \tau \leq 1\) and \(E\) satisfies \((*)\), then there exists \(C(E)>0\) such that \(|\{p+k\in A, p\leq x\}|> \frac{C(E)x}{(\log x)^{2-\tau}}\). If \(2 \in E\), this also holds for odd \(k\). A result on large prime factors of shifted elements: Suppose there is some constant \(B>0\) such that \(\sum_{ p \in E, y \leq p <2y} 1\gg \frac{y}{(\log y)^B}\). Let \(k\neq 0\) be an integer and \(\varepsilon >0\). Then there exists \(C(E)>0\) such that there are at least \(C(E) \frac{x}{(\log x)^{3B}}\) solutions to \(m \in A\), \(m+k=rp\), \(p>x^{3/5 - \varepsilon}\). Moreover, the size of the prime factors of \(m\) can also be restricted: Let \(P(n)\) denote the greatest prime factor of \(n\). Then there are infinitely many solutions to \[ m+k=n,\quad P(m)< \exp\left( (\log m)^{\frac{1}{2}+\varepsilon}\ri…
reviewed by: Christian Elsholtz
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 5620966
zbMATH Keywords: prime factors
zbMATH Keywords: distribution in short interval
zbMATH Keywords: shifted primes
MaRDI profile type: Publication
OpenAlex ID: W2085314164
cites work: Q4885299
cites work: Shifted primes without large prime factors
cites work: The difference between consecutive primes. II
cites work: Analyse de la loi asymptotique de la distribution des nombres premiers généralises
cites work: Primes in arithmetic progressions to large moduli. II
cites work: Q3900124
cites work: Sur la distribution des entiers ayant certaines propriétés
cites work: Additive decomposability of multiplicatively defined sets
cites work: Multiplicative decomposability of shifted sets
cites work: Q2723209
cites work: Q4052222
cites work: On the greatest prime factor of $p-1$ with effective constants
cites work: Q5754222
cites work: Integers without large prime factors
cites work: On the difference between consecutive primes
cites work: Primes of the type φ(x, y)+A where φ is a quadratic form
cites work: Rosser's sieve
cites work: Prime numbers of the form p=m2+n2+1 in short intervals
cites work: The large sieve
cites work: Hilbert's Inequality
cites work: Q4039732
cites work: On the greatest prime factor of integers of the form \(ab+1\)
cites work: Cribler les entiers sans grand facteur premier
cites work: Q3735790
cites work: An elementary method in prime number theory
cites work: Über die Zahlen, deren Primteiler einer gegebenen Menge angehören
DOI: 10.7169/FACM/1229624649
Recommended article: Q4231611
Recommended article: Q5326670
Recommended article: Q4415377
Recommended article: Properties of the Beurling generalized primes
Recommended article: On shifted primes with large prime factors and their products
Recommended article: Zero-density sets containing integers with at most two prime factors
Recommended article: Q3947741
Recommended article: A Hardy-Ramanujan-type inequality for shifted primes and sifted sets
Recommended article: Q3726216
Recommended article: Q5326664
description: scientific article; zbMATH DE number 5620966