A BarbanDavenportHalberstam asymptotic for number fields
Author:
Ethan Smith
Journal:
Proc. Amer. Math. Soc. 138 (2010), 23012309
MSC (2010):
Primary 11N36, 11R44
Published electronically:
March 3, 2010
MathSciNet review:
2607859
Fulltext PDF Free Access
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Abstract: Let be a fixed number field, and assume that is Galois over . Previously, the author showed that when estimating the number of prime ideals with norm congruent to modulo via the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all and all appropriate . In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical BarbanDavenportHalberstam Theorem in the case .
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 M.B. Barban.
On the distribution of primes in arithmetic progressions ``on average''. Dokl. Akad. Nauk SSSR, 5:57, 1964 (Russian).
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 H. Davenport and H. Halberstam.
Primes in arithmetic progressions. Michigan Math. J., 13:485489, 1966. MR 0200257 (34:156)
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 H. Davenport and H. Halberstam.
Corrigendum: ``Primes in arithmetic progression''. Michigan Math. J., 15:505, 1968. MR 0233778 (38:2099)
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 Harold Davenport.
Multiplicative Number Theory. SpringerVerlag, New York, 1980. MR 606931 (82m:10001)
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 Larry Joel Goldstein.
A generalization of the SiegelWalfisz theorem. Trans. Amer. Math. Soc., 149:417429, 1970. MR 0274416 (43:181)
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 Jürgen G. Hinz.
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 H. L. Montgomery.
Primes in arithmetic progressions. Michigan Math. J., 17:3339, 1970. MR 0257005 (41:1660)
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 M. Ram Murty.
Problems in analytic number theory, volume 206 of Graduate Texts in Mathematics. Readings in Mathematics. SpringerVerlag, New York, 2001. MR 1803093 (2001k:11002)
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 Ethan Smith.
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 Robin J. Wilson.
The large sieve in algebraic number fields. Mathematika, 16:189204, 1969. MR 0263774 (41:8374)
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Additional Information
Ethan Smith
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, Michigan 499311295
Email:
ethans@mtu.edu
DOI:
http://dx.doi.org/10.1090/S0002993910103037
Keywords:
Generalized SiegelWalfisz theorem,
BarbanDavenportHalberstam 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.
