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Sums of smooth squares

Part of: Multiplicative number theory Additive number theory; partitions Forms and linear algebraic groups

Published online by Cambridge University Press:  03 December 2009

V. Blomer ,
J. Brüdern  and
R. Dietmann
V. Blomer
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4, USA (email: [email protected])
J. Brüdern
Affiliation:
Universität Stuttgart, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70511 Stuttgart, Germany (email: [email protected])
R. Dietmann
Affiliation:
Universität Stuttgart, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70511 Stuttgart, Germany (email: [email protected])
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Abstract

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Let R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ) of the expected order of magnitude is established, and when θ>365/592, it is shown that R(n,θ)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ>0. This last result is the key to bound R(n,θ) from below.

Keywords

sums of squaressmooth numbersenveloping sieve

MSC classification

Secondary: 11P05: Waring's problem and variants 11N36: Applications of sieve methods 11E25: Sums of squares and representations by other particular quadratic forms
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009 , pp. 1401 - 1441
Copyright
Copyright © Foundation Compositio Mathematica 2009

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