Sums of Two Squares in Short Intervals

Let denote the set of integers representable as a sum of two squares. Since can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that hasmany 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 than expected, and infinitely many intervals containing considerably fewer than expected.