Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T20:08:11.042Z Has data issue: false hasContentIssue false

Sums of Two Squares in Short Intervals

Published online by Cambridge University Press:  20 November 2018

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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Buchstab, A. A., An asymptotic estimate of a number theoretic function. Mat. Sbornik N. S. 44(1937), 12391246.Google Scholar
[2] Friedlander, J. B., Sifting short intervals. Math. Proc. Cambridge Philos. Soc. 91(1982), 915.Google Scholar
[3] Friedlander, J. B., Sifting short intervals. II. Math. Proc. Cambridge Philos. Soc. 92(1982), 381384.Google Scholar
[4] Granville, A., Unexpected irregularities in the distribution of prime numbers. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 388399.Google Scholar
[5] Halberstam, H. and Richert, H.-E., Sieve methods. London Mathematical Soc. Monographs 4, Academic Press, London-New York, 1974.Google Scholar
[6] Hooley, C., On the intervals between numbers that are sums of two squares, IV. J. Reine Angew. Math. 452(1994), 79109.Google Scholar
[7] Iwaniec, H., The half dimensional sieve. Acta Arith. 29(1976), 6995.Google Scholar
[8] Landau, E., Handbuch der Lehre der Verteilung der Primzahlen, Bd. 2. Teubner, Leipzig-Berlin, 1909.Google Scholar
[9] Maier, H., Primes in short intervals. Michigan Math. J. 32(1985), 221225.Google Scholar
[10] Moree, P., On the number of y-smooth natural numbers ≤ x representable as a sum of two integer squares. Manuscripta Math. 80(1993), 199211.Google Scholar
[11] Plaksin, V. A., The distribution of numbers that can be represented as the sum of two squares. Izv. Akad. Nauk SSSR Ser. Mat. 51(1987), 860877.Google Scholar
[12] Plaksin, V. A., Letter to the editors: “The distribution of numbers that can be represented as the sum of two squares” [Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 860–877, 911]. Izv. Akad. Nauk SSSR Ser. Mat. 56(1992), 908909.Google Scholar
[13] Richards, I., On the gaps between numbers which are sums of two squares. Adv. in Math. 46(1982), 12.Google Scholar
[14] Rieger, G. J., Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke II. J. Reine Angew.Math. 217(1965), 200216.Google Scholar
[15] Tenenbaum, G., Introduction to analytic and probabilistic number theory. Cambridge Stud. Adv. Math. 46, Cambridge University Press, Cambridge, 1995.Google Scholar
[16] Wheeler, F. S., Two differential-difference equations arising in number theory. Trans. Amer. Math. Soc. 318(1990), 491523.Google Scholar