On the number of solutions of \(p+h=P_ r\)
type: Item
instance of: scholarly article
title: On the number of solutions of \(p+h=P_ r\)
zbMATH Open document ID: 0659.10056
DOI: 10.1007/BF02570721
author: Jiahai Kan
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174112
review text: Let \(x\) be a sufficiently large positive number, h(\(\neq 0)\) a fixed even number, \(p,p_ 1,...,p_ r\) primes and \(P_ r\) an almost prime with at most r prime factors counted with multiplicity. The work to search for the lower bound for \({\#}\{p: p\leq x\), \(p+h=P_ r\}\) as well as possible is closely connected with the well known Twin Prime Conjecture. In all papers published up to date the order of magnitude for the lower bound is at most \(x \ln^{-2}x\), much smaller than the expected order \(x \ln^{-2}x(\ln \ln x)^{r-1}\). In this paper we show the following: Theorem 1. For \(r\geq 3\), \({\#}\{p: p\leq x\), \(p+h=P_ r\}\gg_{h,r}^ x \ln^{-2}(\ln \ln x)^{r-2};\) Theorem 2. For \(r\geq 3\), \({\#}\{p: p\leq x\), \(p+h=p_ 1...p_{r-2}\) or \(p_ 1...p_{r-1}\) or \(p_ 1...p_ r\}\gg_{h,r}^ x \)ln\({}^{- 2}x(\ln \ln x)^{r-2}.\) Theorem 3. \({\#}\{p: p\leq x\), \(p+h=p_ 1p_ 2\) or \(p_ 1p_ 2p_ 3\}\gg_ h\;x \ln^{-2}x \ln \ln x;\) and miss the presumably correct order of magnitude only by a factor of \(\ln\ln x\).
reviewed by: Jiahai Kan
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 4077365
zbMATH Keywords: almost prime
zbMATH Keywords: twin prime conjecture
zbMATH Keywords: order of magnitude
zbMATH Keywords: lower bound
zbMATH Keywords: weighted sieve method
MaRDI profile type: Publication
cites work: Q4052222
cites work: Primes in arithmetic progressions
cites work: Around the Bombieri-Vinogradov theorem
OpenAlex ID: W40134110
description: scientific article; zbMATH DE number 4077365