The distribution of the number of prime divisors of sums \(a+b\)
type: Item
instance of: scholarly article
title: The distribution of the number of prime divisors of sums \(a+b\)
zbMATH Open document ID: 0646.10042
DOI: 10.1016/0022-314X(88)90094-7
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(A,B\subset [1,x]\) be sets of integers and let \(F(z)\) denote the frequency of those pairs \((a,b)\), \(a\in A\), \(b\in B\) that satisfy \[ \omega (a+b)-\log \log x<z\sqrt{\log \log x}, \] where \(\omega\) denotes the number of prime divisors. \textit{P. Erdős}, \textit{H. Maier} and \textit{A. Sárközy} [Trans. Am. Math. Soc. 302, 269--280 (1987; Zbl 0617.10038)] showed that for large sets \(F(z)\) approximates the standard normal distribution \(G(z)\). The authors prove the following effective generalization of this result. Assume that \[ \vert B\vert >x(\log x)^{-c}\quad \text{and}\quad \vert A\vert >x \exp (-\lambda (\log \log x)^{1/2} \log \log x); \] then we have \[ F(z)-G(z)\ll \lambda +(\log \log x)^{\mp\varepsilon}, \] where the implied constant depends on \(c\) and \(\varepsilon\).
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N60
zbMATH DE Number: 4053748
zbMATH Keywords: frequency
zbMATH Keywords: number of prime divisors
zbMATH Keywords: normal distribution
author: Peter D. T. A. Elliott
author: András Sárközy
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
cites work: Q3867876
cites work: Q3347966
cites work: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law
description: scientific article; zbMATH DE number 4053748