On locally repeated values of certain arithmetic functions. II
type: Item
instance of: scholarly article
title: On locally repeated values of certain arithmetic functions. II
zbMATH Open document ID: 0609.10034
DOI: 10.1007/BF01956329
author: Carl Pomerance
author: András Sárközy
author: P. Erdős
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: [For part I, cf. J. Number Theory 21, 319-332 (1985; Zbl 0574.10012).] It seems reasonable, in view of the Erdős-Kac theorem, to conjecture that the number of \(n\leq x\) with \(\nu(n)=\nu(n+1)\) is of exact order \(x(\log\log x)^-\). The principal result of the paper is that there is a constant c such that \(|\nu(n)-\nu(n+1)| \leq c\) has at least order \(x(\log\log x)^-\) solutions \(n\leq x\). It is stated that \(c=3\) is admissible. The proof uses the Erdős-Kac theorem to produce many pairs \(a,b\) with \(\nu(a) = \nu(b)\). One then takes \(a_0,b_0\) to solve \(ab_0-ba_0=1\), and uses the small sieve to find integers \(m\) for which \(\nu(am+a_0)\), \(\nu(bm+b_0)\) are both small. One may then take \(n=b(am+a_0)\). The reviewer [Mathematika 31, 141-149 (1984; Zbl 0529.10040)] showed by a different approach that the related equation \(d(n)=d(n+1)\) (where \(d(n)\) is the divisor function) has infinitely many solutions. It seems likely that one could handle \(\nu(n) = \nu(n+1)\) the same way, but as yet there seems no hope of ''hybridizing'' the two methods so as to solve the original conjecture. In the final section of the paper, the number of solutions \(n\leq x\) of the equation \(\phi(n) = \phi(n+1)\), (where \(\phi(n)\) is Euler's function) is considered, and shown to be at most \(x/\exp((\log x)^{1/3})\). The proof is closely related to \textit{C. Pomerance}'s treatment [J. Reine Angew. Math. 325, 183-188 (1981; Zbl 0448.10007)] of amicable numbers.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 3985336
zbMATH Keywords: sieve method
zbMATH Keywords: Erdős-Kac theorem
zbMATH Keywords: Euler's function
Wikidata QID: Q105890424
MaRDI profile type: Publication
cites work: Q5803354
cites work: Q3867876
cites work: On the integers having exactly \(k\) prime factors
cites work: Q4052222
cites work: The divisor function at consecutive integers
cites work: On the distribution of amicable numbers. II.
cites work: Q5818563
full work available at URL: https://doi.org/10.1007/bf01956329
OpenAlex ID: W2002004779
description: scientific article; zbMATH DE number 3985336