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Sums of Two Squares in Short Intervals

Published online by Cambridge University Press:  20 November 2018

Antal Balog  and
Trevor D. Wooley
Antal Balog
Affiliation:
Mathematical Institute, Budapest 1364, Hungary email: [email protected]
Trevor D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA email: [email protected]
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Abstract

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

Keywords

11N3611N3711N25sums of two squaressievesshort intervalssmooth numbers
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 4 , 01 August 2000 , pp. 673 - 694
Copyright
Copyright © Canadian Mathematical Society 2000

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