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

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