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The number of primes representable as the sum of two square-free squares

Published online by Cambridge University Press:  20 January 2009

M. Keates
M. Keates
Affiliation:
Inverness Technical College, Longman Road, Inverness
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In this paper an asymptotic formula is obtained for the number of primes representable as the sum of two square-free squares. The precise result is:

Theorem 1.Let N(x) be the number of primes not exceeding x represented by the quadratic formy2 + z2, where y and z are square-free. Let w be a fixed arbitrarily large number. Then

where

and

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 1 , March 1976 , pp. 23 - 27
Copyright
Copyright © Edinburgh Mathematical Society 1976

References

(1) Estermann, T., Introduction to modern prime number theory (Cambridge University Press, 1961).Google Scholar
(2) Keates, M., Sums of squares of square-free numbers, Ph.D. thesis (University of Wales, 1971).Google Scholar
(3) Matthews, G. B., Theory of numbers, Part I (Cambridge University Press. 1892).Google Scholar
(4) de la Vallée Poussin, C. J., Recherches analytiques sur la théorie des nombrès. Troisieme partie, Annates de la Soc. Sci. de Bruxelles 20 (1896), 363.Google Scholar