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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The distribution of prime ideals of imaginary quadratic fields


Authors: G. Harman, A. Kumchev and P. A. Lewis
Journal: Trans. Amer. Math. Soc. 356 (2004), 599-620
MSC (2000): Primary 11R44; Secondary 11N32, 11N36, 11N42.
Published electronically: September 22, 2003
MathSciNet review: 2022713
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $Q(x, y)$ be a primitive positive definite quadratic form with integer coefficients. Then, for all $(s, t)\in \mathbb R^2$ there exist $(m, n) \in \mathbb Z^2$ such that $Q(m, n)$ is prime and

\begin{displaymath}Q(m - s, n - t) \ll Q(s, t)^{0.53} + 1. \end{displaymath}

This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.


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Additional Information

G. Harman
Affiliation: Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom
Email: G.Harman@rhul.ac.uk

A. Kumchev
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: kumchev@math.toronto.edu

P. A. Lewis
Affiliation: School of Mathematics, Cardiff University, P.O. Box 926, Cardiff CF24 4YH, Wales, United Kingdom
Email: LewisPA3@Cardiff.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03104-0
PII: S 0002-9947(03)03104-0
Received by editor(s): January 11, 2002
Received by editor(s) in revised form: April 22, 2002
Published electronically: September 22, 2003
Additional Notes: The second author was partially supported by NSF Grant DMS 9970455 and NSERC Grant A5123.
The third author was supported by an EPSRC Research Studentship.
Article copyright: © Copyright 2003 American Mathematical Society