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Binary linear forms as sums of two squares

Part of: Diophantine equations Multiplicative number theory

Published online by Cambridge University Press:  01 November 2008

R. de la Bretèche  and
T. D. Browning
R. de la Bretèche
Affiliation:
Institut de Mathématiques de Jussieu, Université Paris 7 Denis Diderot, Case Postale 7012, 2 Place Jussieu, F-75251 Paris cedex 05, France (email: breteche@math.jussieu.fr)
T. D. Browning
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK (email: t.d.browning@bristol.ac.uk)
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Abstract

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We revisit recent work of Heath-Brown on the average order of the quantity r(L1(x))⋯r(L4(x)), for suitable binary linear forms L1,…,L4, as x=(x1,x2) ranges over quite general regions in ℤ2. In addition to improving the error term in Heath-Brown’s estimate, we generalise his result to cover a wider class of linear forms.

Keywords

rational points of bounded heightbinary formsdivisor problemManin conjecturesums of two squaresuniform asymptotic formula

MSC classification

Secondary: 11N37: Asymptotic results on arithmetic functions 11D25: Cubic and quartic equations 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 6 , November 2008 , pp. 1375 - 1402
Copyright
Copyright © Foundation Compositio Mathematica 2008