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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.

Let $\mathcal{S}$ denote the set of integers representable as a sum of two squares. Since $\mathcal{S}$ can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that $\mathcal{S}$ has many properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of $\mathcal{S}$ than expected, and infinitely many intervals containing considerably fewer than expected.

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Binary linear forms as sums of two squares

Sums of Two Squares in Short Intervals

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Binary linear forms as sums of two squares

Sums of Two Squares in Short Intervals