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 | References | Similar Articles | Additional Information

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 Barban-Davenport-Halberstam Theorem in the case .

**1.**M.B. Barban.

On the distribution of primes in arithmetic progressions ``on average''.*Dokl. Akad. Nauk SSSR*, 5:5-7, 1964

(Russian).**2.**H. Davenport and H. Halberstam,*Primes in arithmetic progressions*, Michigan Math. J.**13**(1966), 485–489. MR**0200257****3.**H. Davenport and H. Halberstam,*Corrigendum: “Primes in arithmetic progression”*, Michigan Math. J.**15**(1968), 505. MR**0233778****4.**Harold Davenport,*Multiplicative number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR**606931****5.**Larry Joel Goldstein,*A generalization of the Siegel-Walfisz theorem*, Trans. Amer. Math. Soc.**149**(1970), 417–429. MR**0274416**, 10.1090/S0002-9947-1970-0274416-6**6.**Jürgen G. Hinz,*On the theorem of Barban and Davenport-Halberstam in algebraic number fields*, J. Number Theory**13**(1981), no. 4, 463–484. MR**642922**, 10.1016/0022-314X(81)90038-X**7.**Christopher Hooley,*On the Barban-Davenport-Halberstam theorem. I*, J. Reine Angew. Math.**274/275**(1975), 206–223. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III. MR**0382202****8.**Serge Lang,*Algebraic number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR**1282723****9.**A. F. Lavrik,*On the twin prime hypothesis of the theory of primes by the method of I. M. Vinogradov*, Soviet Math. Dokl.**1**(1960), 700–702. MR**0157955****10.**H. L. Montgomery,*Primes in arithmetic progressions*, Michigan Math. J.**17**(1970), 33–39. MR**0257005****11.**M. Ram Murty,*Problems in analytic number theory*, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR**1803093****12.**Ethan Smith,*A generalization of the Barban-Davenport-Halberstam theorem to number fields*, J. Number Theory**129**(2009), no. 11, 2735–2742. MR**2549528**, 10.1016/j.jnt.2009.05.005**13.**Robin J. Wilson,*The large sieve in algebraic number fields*, Mathematika**16**(1969), 189–204. MR**0263774**

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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:
https://doi.org/10.1090/S0002-9939-10-10303-7

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.