Sums of two squares in short intervals

Let S denote the set of integers representable as a sum of two squares. Sin ce S can be described as the unsifted elements of a sieving process of posi tive dimension, it is to be expected that S has many properties in common w ith the set of prime numbers. In this paper we exhibit "unexpected irregula rities" in the distribution of sums of two squares in short intervals, a ph enomenon analogous to that discovered by Maier, over a decade ago, in the d istribution of prime numbers. To be precise, we show that there are infinit ely many short intervals containing considerably more elements of S than ex pected, and infinitely many intervals containing considerably fewer than ex pected.