Sums of Two Squares in Short Intervals

http://dx.doi.org/10.4153/CJM-2000-029-6
Canad. J. Math. 52(2000), 673-694

  Published:2000-08-01
 Printed: Aug 2000

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