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On the sum of a square and a square of a prime

Published online by Cambridge University Press:  26 October 2001

STEPHAN DANIEL
STEPHAN DANIEL
Affiliation:
School of Mathematics, Cardiff University, Cardiff CF2 4YH, Wales. e-mail: [email protected]
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Abstract

Let r1(n) denote the number of representations of the positive integer n as the sum of a square of a positive integer and the square of a positive prime number. We prove an asymptotic evaluation for [sum ]n[les ]xr1(n)2, as x → ∞, thereby improving upon a O-result of Rieger [7]. We further prove an asymptotic formula for the number of positive integers n [les ] x with r1(n) [ges ] 1, which answers a question stated at the end of [7]. Our result in particular shows that for almost all integers represented as the sum of a square and a square of a prime, the representation is unique.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 131 , Issue 1 , July 2001 , pp. 1 - 22
Copyright
2001 Cambridge Philosophical Society

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