Mean square of the remainder term in the Dirichlet divisor problem
type: Item
instance of: scholarly article
title: Mean square of the remainder term in the Dirichlet divisor problem
zbMATH Open document ID: 0844.11059
author: Yuk-Kam Lau
author: Kai-Man Tsang
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93632
full work available at URL: jtnb7-1.html
review text: If \(\Delta (x)\) is the error term in the Dirichlet divisor problem, then it was shown by \textit{K.-C. Tong} [Acta Math. Sin. 6, 515-541 (1956; Zbl 0075.25003)]\ that \(\int^X_2 \Delta (x)^2 dx= cX^{3/2}+ F(X)\) for an appropriate constant \(c\), where \(F(X)\ll X\log^5 X\). It has been conjectured that \(F(X)\ll X^{3/4+ \varepsilon}\) for any \(\varepsilon>0\), which would imply that \(\Delta (x)\ll x^{1/4+ \varepsilon}\). However it is shown in the present paper that \[ \int^X_2 F(x) dx=- (8\pi^2 )^{-1} X^2\log^2 X+ c_1 X^2\log X+ O(X^2) \] for a certain constant \(c_1\). Thus, in particular, \(F(X)= \Omega_- (X\log^2 x)\). The proof gives a careful analysis of \(F(X)\) which ultimately depends on the asymptotic formula for \(\sum_{m\leq x} d(m) d(m+ h)\).
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 858015
zbMATH Keywords: mean square
zbMATH Keywords: error term
zbMATH Keywords: Dirichlet divisor problem
MaRDI profile type: Publication
OpenAlex ID: W1985863245
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DOI: 10.5802/JTNB.131
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description: scientific article; zbMATH DE number 858015