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Gaussian primes in narrow sectors

Part of:Multiplicative number theory

Published online by Cambridge University Press:  26 February 2010

Glyn Harman and
Philip Lewis
Glyn Harman
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 OEX
Philip Lewis
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, P.O. Box 926, Cardiff CF12 4YH.

Extract

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The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more generally, prime ideals in algebraic number fields). One consequence of our main theorem (Theorem 2 below) is the following result on rational primes.

MSC classification

Secondary:11N36: Applications of sieve methods

Information

Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001, pp. 119 - 135
Copyright
Copyright © University College London 2001

References

1.Ankeny, N. C.. Representations of primes by quadratic forms. American J. Math., 74 (1952). 913–919.CrossRefGoogle Scholar
2.Baker, R. C.. Diophantine Inequalities. LMS Monographs (New Series), 1, (Clarendon Press, Oxford, 1986).Google Scholar
3.Baker, R. C., Harman, G. and Pintz, J.. The difference between consecutive primes II. Proc. London Math. Soc, (3), 83 (2001), 532–562.Google Scholar
4.Balog, A.. On the distribution of pϑ mod 1. Ada Math. Hungar. 45 (1985), 179–199.CrossRefGoogle Scholar
5.Coleman, M. D.. The distribution of points at which binary quadratic forms are prime. Proc. London Math. Soc, (3), 61 (1990), 433–456.CrossRefGoogle Scholar
6.Coleman, M. D.. A zero-free region for the Hecke L-functions. Mathematika, 37 (1990), 287–304.Google Scholar
7.Coleman, M. D.. The Rosser-Iwaniec sieve in number fields. Acta Arithmetica, 65 (1993), 53–83.Google Scholar
8.Coleman, M. D.. Relative norms of prime ideals in small regions. Mathematika, 43 (1996), 40– 62.Google Scholar
9.Harman, G.. On the distribution of ap moduls one. J. London Math. Soc. (2), 27 (1983), 9– 18.Google Scholar
10.Harman, G.. On the distribution of √p modulo one. Mathematika, 30 (1983), 104–116.CrossRefGoogle Scholar
11.Harman, G.. On the distribution of αp modulo one II. Proc. London Math. Soc. (3), 72 (1996). 241–260.CrossRefGoogle Scholar
12.Hecke, E.. Eine neue Art von Zetafunctionen und ihre Beziehung zur Verteilung der Primzahlen I, II. Math. Zeit., 1 (1918), 357–376; 6 (1920), 11–51.CrossRefGoogle Scholar
13.Kubilius, J. P.. On a problem in the n-dimensional analytic theory of numbers. Viliniaus Valst. Univ. Mokslo dardai Fiz. Chem. Moksly Ser., 4 (1955), 5–43.Google Scholar
14.Matsui, H.. A bound for the least Gaussian prime ω with α< arg(ω<β. Arch. Math., 74 (2000), 423–4321.CrossRefGoogle Scholar
15.Ricci, S.. Local Distribution of Primes (PhD Thesis, University of Michigan, 1976).Google Scholar
16.Titchmarsh, E. C.. The theory of the Riematm Zeta-function (second edition revised by Heath-Brown, D. R.) (Clarendon Press, Oxford, 1986).Google Scholar