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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Barban-Davenport-Halberstam asymptotic for number fields


Author: Ethan Smith
Journal: Proc. Amer. Math. Soc. 138 (2010), 2301-2309
MSC (2010): Primary 11N36, 11R44
Published electronically: March 3, 2010
MathSciNet review: 2607859
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Abstract: Let $ K$ be a fixed number field, and assume that $ K$ is Galois over $ \mathbb{Q}$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $ a$ modulo $ q$ via the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all $ q\le Q$ and all appropriate $ a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $ K=\mathbb{Q}$.


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Additional Information

Ethan Smith
Affiliation: Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, Michigan 49931-1295
Email: ethans@mtu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10303-7
PII: S 0002-9939(10)10303-7
Keywords: Generalized Siegel-Walfisz theorem, Barban-Davenport-Halberstam Theorem
Received by editor(s): July 28, 2009
Received by editor(s) in revised form: October 15, 2009, and October 30, 2009
Published electronically: March 3, 2010
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.