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Binary linear forms as sums of two squares. (English) Zbl 1234.11132
The authors study recent work of D. R. Heath-Brown [Number theory and algebraic geometry. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 303, 133–176 (2003; Zbl 1161.11387)] on the average order of the quantity \(r(L_{1}(\mathbf x))\cdots r(L_{4}(\mathbf x))\), for suitable binary linear forms \(L_{1},\ldots ,L_{4}\), as \(\mathbf x=(x_{1},x_{2})\) ranges over quite general regions in \(\mathbb Z^{2}\). First they improve the error term in Heath-Brown’s estimate (see the paper for details), and then they generalise his result to cover a wider class of linear forms.

11N37 Asymptotic results on arithmetic functions
11D25 Cubic and quartic Diophantine equations
11N25 Distribution of integers with specified multiplicative constraints
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