Estimates in the general divisor problem
type: Item
instance of: scholarly article
title: Estimates in the general divisor problem
zbMATH Open document ID: 0776.11057
DOI: 10.1007/BF02941626
author: Ekkehard Krätzel
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1993-08-31T00:00:00Z
review text: Three theorems are proved, which involve upper bounds for \[ \Delta({\mathbf a};x)=\sum_{n_ 1^{a_ 1} n^{a_ 2}_ 2\cdots n^{a_ p}_ p\leq x} 1-\sum^ p_{i=1}\left(\prod^ p_{{j=1\atop j\neq i}}\zeta\bigl({a_ j\over a_ i}\bigr)\right)x^{1/a_ i} \] if \(1\leq a_ 1< a_ 2<\cdots< a_ p\) are given integers, and \(n_ j\) \((\geq 1)\) are integer variables. If some of the \(a_ j\)'s are equal, then the appropriate limits in the second sum above are to be taken. The function \(\Delta({\mathbf a}; x)\) may be considered as the error term in the general divisor problem, and there exists a considerable literature on this subject [see e.g. the reviewer's paper in J. Number Theory 27, 73-91 (1987; Zbl 0619.10046) for omega-results]. The author's first theorem says that, if \(p\geq 2\) and \((p-2)a_ p< 2a_ 1+ a_ 2+\cdots +a_{p- 1}\), then \[ \Delta({\mathbf a};x)\ll x^{(p-1)/(2a_ 1+a_ 2+\cdots +a_ p)}\log^ p x. \] The formulation of the remaining theorems is more complicated, and for this reason they will not be stated here. The proofs involve delicate transformations of multiple exponential sums and the use of the technique of exponent pairs.
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 404315
zbMATH Keywords: error term
zbMATH Keywords: general divisor problem
zbMATH Keywords: transformations of multiple exponential sums
zbMATH Keywords: exponent pairs
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
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cites work: New omega results in a weighted divisor problem
cites work: The general divisor problem
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cites work: Q3994503
cites work: On the average number of direct factors of a finite Abelian group
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description: scientific article; zbMATH DE number 404315