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On the representation of an integer as the sum of a large and a small square

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  26 February 2010

R. R. Hall
R. R. Hall
Affiliation:
Department of Mathematics, University of York. Heslington, York YO10 5DD, U.K.
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Extract

We continue our study of the different representations of an integer n as a sum of two squares initiated in our final paper written with Paul Erdős [1]. We let r(n) denote the number of representations n = a2 + b2 counted in the usual way, that is, with regard to both the order and sign of a and b. We have

where x is the non-principal Dirichlet character (mod 4); moreover, r(n)≥0 if and only if n has no prime factor p ≡ 3 (mod 4) with odd exponent. We define the function b(n) on the sequence of representable numbers as the least possible value of |b|—for example, we have b(13) = 2,b(25) = 0, b(65) = 1—and we write

here and throughout the paper the star denotes that the sum is restricted to the representable integers. The problem considered here is to find an asymptotic formula for this sum, or, less ambitiously, to determine the order of magnitude of the function B(x).

MSC classification

Secondary: 11E25: Sums of squares and representations by other particular quadratic forms
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 245 - 251
Copyright
Copyright © University College London 2002

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References

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