Hooley's \(\Delta _ r\)-functions when r is large
type: Item
instance of: scholarly article
title: Hooley's \(\Delta _ r\)-functions when r is large
zbMATH Open document ID: 0591.10039
author: B. George
published in: Michigan Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: In an important paper [Proc. Lond. Math. Soc., III. Ser. 38, 115-151 (1979; Zbl 0394.10027)], \textit{C. Hooley} introduced the functions \[ \Delta_ r(n)=\max_{u_ 1,...,u_{r-1}}card\{d_ 1,...,d_{r- 1}:\quad d_ 1...d_{r-1}| n,\quad u_ i<d_ i\leq eu_ i\quad (i=1,...,r-1)\}, \] and showed that estimates for \(\Delta_ r(n)\) have potential applications to various problems in additive number theory and diophantine approximation. Using Fourier techniques, Hooley established bounds of the type \((*)\quad \sum_{n\leq x}\Delta_ r(n) \ll x (\log x)^{\xi}\) with certain positive constants \(\xi =\xi (r).\) By estimating more carefully a multiple integral arising in Hooley's argument, the author improves on Hooley's values for the exponents in (*); for example, he shows that if \(4\leq r\leq 13\), then (*) holds with \(\xi =(r-1)^{3/2}/(r+\sqrt{r-1}).\) Remark: In the meantime, much stronger estimates have been obtained by the author and \textit{G. Tenenbaum} in a paper to appear in Compos. Math.).
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3948398
zbMATH Keywords: distribution of divisors
zbMATH Keywords: Hooley delta-functions
reviewed by: Adolf Hildebrand
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003293
OpenAlex ID: W2066700481
DOI: 10.1307/MMJ/1029003293
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description: scientific article; zbMATH DE number 3948398