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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.
Posted:
September 22, 2003
MathSciNet review:
2022713
Full-text PDF Free Access
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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.
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- 1.
- R. C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. (3) 72 (1996), 261-280. 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. 1-54. MR 99g:11121
- 3.
- -, The difference between consecutive primes II, Proc. London Math. Soc. (3) 83 (2001), 532-562. 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), 433-456. MR 91j:11077
- 5.
- -, A zero-free region for the Hecke
 -functions, Mathematika 37 (1990), 287-304. MR 92c:11131
- 6.
- -, The Rosser-Iwaniec sieve in number fields, with an application, Acta Arith. 65 (1993), 53-83. MR 94h:11086
- 7.
- -, Relative norms of prime ideals in small regions, Mathematika 43 (1996), 40-62. MR 97f:11094
- 8.
- G. Harman, On the distribution of
 modulo one, J. London Math. Soc. (2) 27 (1983), 9-18. MR 84d:10044
- 9.
- -, On the distribution of
 modulo one II, Proc. London Math. Soc. (3) 72 (1996), 241-260. MR 96k:11089
- 10.
- G. Harman and P. A. Lewis, Gaussian primes in narrow sectors, Mathematika, to appear.
- 11.
- D. R. Heath-Brown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22-63. MR 89i:11099
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- D. R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Invent. Math. 55 (1979), 49-69. MR 81h:10064
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- M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164-170. MR 45:1856
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- H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer-Verlag, Berlin-New York, 1971. MR 49:2616
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- W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Monografie Matematyczne 57, PWN--Polish Scientific Publishers, Warsaw, 1974. MR 50:268
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- H. Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z. 72 (1959/60), 192-204. MR 22:7982
- 17.
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- 18.
- N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), 179-210. 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/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
Posted:
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
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