On the average value for the number of divisors of numbers of form \(ab+1\)
type: Item
instance of: scholarly article
title: On the average value for the number of divisors of numbers of form \(ab+1\)
zbMATH Open document ID: 0841.11047
DOI: 10.1007/BF01874287
published in: Acta Mathematica Hungarica
publication date: 1996-07-16T00:00:00Z
review text: Let \(A\), \(B\) be two sets of distinct integers in the interval \([1,x ]\). Various authors have studied the prime factor or divisibility structure of the sums \(a+b\) and, more recently, \(ab+1\), where \(a\in A\), \(b\in B\). In [Ill. J. Math. 38, No. 1, 1-18 (1994; Zbl 0793.11025)], \textit{A. Sárközy} and \textit{C. L. Stewart} obtained a lower bound for \((|A||B|)^{-1} \sum_{a\in A} \sum_{b\in B} \tau(a+ b)\). The object of this paper is to study the corresponding quantity with summand \(\tau (ab+1)\). However the undue influence of any small prime factors of \(ab+1\) renders the establishment of an asymptotic formula impossible in general, as the author illustrates by means of examples of sets \(A\), \(B\). To avoid this difficulty, he considers instead the sum \[ T_K=( |A||B|)^{-1} \sum_{a\in A} \sum_{b\in B} \tau_K (ab+1) \] where \(\tau_K (n)\) denotes the number of positive divisors \(d\) of \(n\) such that every prime factor of \(d\) exceeds \(K\). He proves the expected result \[ T_K\sim \prod_{p\leq K} (1- {\textstyle {1\over p}}) \log x \] when \(\log K\leq (\log x)^{1\over 2}\), but \(K\) is not too small in comparison with \(U= x^2/|A||B|= o(\log x)\) in a sense that is made explicit.
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 755504
zbMATH Keywords: number of divisors
zbMATH Keywords: divisors with no small prime factor
zbMATH Keywords: asymptotic formula
author: András Sárközy
reviewed by: Eira J. Scourfield
MaRDI profile type: Publication
cites work: The distribution of the number of prime divisors of sums \(a+b\)
cites work: The large sieve
cites work: Topics in multiplicative number theory
cites work: Facteurs Premiers de Sommes D'Entiers
full work available at URL: https://doi.org/10.1007/bf01874287
OpenAlex ID: W2084295990
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description: scientific article; zbMATH DE number 755504