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), 599620
MSC (2000):
Primary 11R44; Secondary 11N32, 11N36, 11N42.
Published electronically:
September 22, 2003
MathSciNet review:
2022713
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Let be a primitive positive definite quadratic form with integer coefficients. Then, for all there exist such that is prime and
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.
 1.
R.
C. Baker and G.
Harman, The difference between consecutive primes, Proc.
London Math. Soc. (3) 72 (1996), no. 2,
261–280. MR 1367079
(96k:11111), http://dx.doi.org/10.1112/plms/s372.2.261
 2.
R.
C. Baker, G.
Harman, and J.
Pintz, The exceptional set for Goldbach’s problem in short
intervals, Sieve methods, exponential sums, and their applications in
number theory (Cardiff, 1995) London Math. Soc. Lecture Note Ser.,
vol. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 1–54.
MR
1635718 (99g:11121), http://dx.doi.org/10.1017/CBO9780511526091.004
 3.
R.
C. Baker, G.
Harman, and J.
Pintz, The difference between consecutive primes. II, Proc.
London Math. Soc. (3) 83 (2001), no. 3,
532–562. MR 1851081
(2002f:11125), http://dx.doi.org/10.1112/plms/83.3.532
 4.
M.
D. Coleman, The distribution of points at which binary quadratic
forms are prime, Proc. London Math. Soc. (3) 61
(1990), no. 3, 433–456. MR 1069510
(91j:11077), http://dx.doi.org/10.1112/plms/s361.3.433
 5.
M.
D. Coleman, A zerofree region for the Hecke
𝐿functions, Mathematika 37 (1990),
no. 2, 287–304. MR 1099777
(92c:11131), http://dx.doi.org/10.1112/S0025579300013000
 6.
M.
D. Coleman, The RosserIwaniec sieve in number fields, with an
application, Acta Arith. 65 (1993), no. 1,
53–83. MR
1239243 (94h:11086)
 7.
M.
D. Coleman, Relative norms of prime ideals in small regions,
Mathematika 43 (1996), no. 1, 40–62. MR 1401707
(97f:11094), http://dx.doi.org/10.1112/S002557930001158X
 8.
Glyn
Harman, On the distribution of 𝛼𝑝 modulo one,
J. London Math. Soc. (2) 27 (1983), no. 1,
9–18. MR
686496 (84d:10044), http://dx.doi.org/10.1112/jlms/s227.1.9
 9.
Glyn
Harman, On the distribution of 𝛼𝑝 modulo one.
II, Proc. London Math. Soc. (3) 72 (1996),
no. 2, 241–260. MR 1367078
(96k:11089), http://dx.doi.org/10.1112/plms/s372.2.241
 10.
G. Harman and P. A. Lewis, Gaussian primes in narrow sectors, Mathematika, to appear.
 11.
D.
R. HeathBrown, The number of primes in a short interval, J.
Reine Angew. Math. 389 (1988), 22–63. MR 953665
(89i:11099), http://dx.doi.org/10.1515/crll.1988.389.22
 12.
D.
R. HeathBrown and H.
Iwaniec, On the difference between consecutive primes, Invent.
Math. 55 (1979), no. 1, 49–69. MR 553995
(81h:10064), http://dx.doi.org/10.1007/BF02139702
 13.
M.
N. Huxley, On the difference between consecutive primes,
Invent. Math. 15 (1972), 164–170. MR 0292774
(45 #1856)
 14.
Hugh
L. Montgomery, Topics in multiplicative number theory, Lecture
Notes in Mathematics, Vol. 227, SpringerVerlag, BerlinNew York, 1971. MR 0337847
(49 #2616)
 15.
Władysław
Narkiewicz, Elementary and analytic theory of algebraic
numbers, PWN—Polish Scientific Publishers, Warsaw, 1974.
Monografie Matematyczne, Tom 57. MR 0347767
(50 #268)
 16.
Hans
Rademacher, On the PhragménLindelöf theorem and some
applications, Math. Z 72 (1959/1960), 192–204.
MR
0117200 (22 #7982)
 17.
E.
C. Titchmarsh, The theory of the Riemann zetafunction, 2nd
ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited
and with a preface by D. R. HeathBrown. MR 882550
(88c:11049)
 18.
N.
Watt, Kloosterman sums and a mean value for Dirichlet
polynomials, J. Number Theory 53 (1995), no. 1,
179–210. MR 1344840
(96f:11109), http://dx.doi.org/10.1006/jnth.1995.1086
 1.
 R. C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. (3) 72 (1996), 261280. MR 96k:11111
 2.
 R. C. Baker, G. Harman and J. Pintz, The exceptional set for Goldbach's problem in short intervals, in ``Sieve Methods, Exponential Sums and their Applications in Number Theory'', London Math. Soc. Lecture Notes 237, Cambridge University Press, 1997, pp. 154. MR 99g:11121
 3.
 , The difference between consecutive primes II, Proc. London Math. Soc. (3) 83 (2001), 532562. MR 2002f:11125
 4.
 M. D. Coleman, The distribution of points at which binary quadratic forms are prime, Proc. London Math. Soc. (3) 61 (1990), 433456. MR 91j:11077
 5.
 , A zerofree region for the Hecke functions, Mathematika 37 (1990), 287304. MR 92c:11131
 6.
 , The RosserIwaniec sieve in number fields, with an application, Acta Arith. 65 (1993), 5383. MR 94h:11086
 7.
 , Relative norms of prime ideals in small regions, Mathematika 43 (1996), 4062. MR 97f:11094
 8.
 G. Harman, On the distribution of modulo one, J. London Math. Soc. (2) 27 (1983), 918. MR 84d:10044
 9.
 , On the distribution of modulo one II, Proc. London Math. Soc. (3) 72 (1996), 241260. MR 96k:11089
 10.
 G. Harman and P. A. Lewis, Gaussian primes in narrow sectors, Mathematika, to appear.
 11.
 D. R. HeathBrown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 2263. MR 89i:11099
 12.
 D. R. HeathBrown and H. Iwaniec, On the difference between consecutive primes, Invent. Math. 55 (1979), 4969. MR 81h:10064
 13.
 M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164170. MR 45:1856
 14.
 H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, SpringerVerlag, BerlinNew York, 1971. MR 49:2616
 15.
 W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Monografie Matematyczne 57, PWNPolish Scientific Publishers, Warsaw, 1974. MR 50:268
 16.
 H. Rademacher, On the PhragménLindelöf theorem and some applications, Math. Z. 72 (1959/60), 192204. MR 22:7982
 17.
 E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, Second ed., edited and with a preface by D. R. HeathBrown, The Clarendeon Press, Oxford University Press, New York, 1986. MR 88c:11049
 18.
 N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), 179210. MR 96f:11109
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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/S0002994703031040
PII:
S 00029947(03)031040
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
