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# Sums of Two Squares in Short Intervals

Published:2000-08-01
Printed: Aug 2000
• Antal Balog
• Trevor D. Wooley
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## Abstract

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.
 Keywords: sums of two squares, sieves, short intervals, smooth numbers
 MSC Classifications: 11N36 - Applications of sieve methods 11N37 - Asymptotic results on arithmetic functions 11N25 - Distribution of integers with specified multiplicative constraints

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