Binary linear forms as sums of two squares

TL;DR

This paper improves and generalizes Heath-Brown's results on the average behavior of products of representation counts of integers as sums of two squares, for binary linear forms over broad regions.

Contribution

It enhances the error term in Heath-Brown's estimate and extends the applicability of his results to more general settings.

Findings

Improved error bounds in average order estimates

Extended the range of binary linear forms covered

Generalized previous results to broader regions

Abstract

We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in Heath-Brown's estimate we generalise his result quite extensively.