Sums of Two Squares in Short Intervals

Balog, Antal ; Wooley, Trevor D.

Canadian journal of mathematics, Tome 52 (2000) no. 4, pp. 673-694

Résumé

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.

Détail

DOI : 10.4153/CJM-2000-029-6

Mots-clés : 11N36, 11N37, 11N25, sums of two squares, sieves, short intervals, smooth numbers

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