Skip to main content Accessibility help
×
×
Home

Gaussian primes in narrow sectors

  • Glyn Harman (a1) and Philip Lewis (a2)
Extract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Gaussian primes in narrow sectors
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Gaussian primes in narrow sectors
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Gaussian primes in narrow sectors
      Available formats
      ×
Copyright
References
Hide All
1.Ankeny, N. C.. Representations of primes by quadratic forms. American J. Math., 74 (1952). 913919.
2.Baker, R. C.. Diophantine Inequalities. LMS Monographs (New Series), 1, (Clarendon Press, Oxford, 1986).
3.Baker, R. C., Harman, G. and Pintz, J.. The difference between consecutive primes II. Proc. London Math. Soc, (3), 83 (2001), 532562.
4.Balog, A.. On the distribution of pϑ mod 1. Ada Math. Hungar. 45 (1985), 179199.
5.Coleman, M. D.. The distribution of points at which binary quadratic forms are prime. Proc. London Math. Soc, (3), 61 (1990), 433456.
6.Coleman, M. D.. A zero-free region for the Hecke L-functions. Mathematika, 37 (1990), 287304.
7.Coleman, M. D.. The Rosser-Iwaniec sieve in number fields. Acta Arithmetica, 65 (1993), 5383.
8.Coleman, M. D.. Relative norms of prime ideals in small regions. Mathematika, 43 (1996), 4062.
9.Harman, G.. On the distribution of ap moduls one. J. London Math. Soc. (2), 27 (1983), 918.
10.Harman, G.. On the distribution of √p modulo one. Mathematika, 30 (1983), 104116.
11.Harman, G.. On the distribution of αp modulo one II. Proc. London Math. Soc. (3), 72 (1996). 241260.
12.Hecke, E.. Eine neue Art von Zetafunctionen und ihre Beziehung zur Verteilung der Primzahlen I, II. Math. Zeit., 1 (1918), 357376; 6 (1920), 11–51.
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), 543.
14.Matsui, H.. A bound for the least Gaussian prime ω with α< arg(ω<β. Arch. Math., 74 (2000), 4234321.
15.Ricci, S.. Local Distribution of Primes (PhD Thesis, University of Michigan, 1976).
16.Titchmarsh, E. C.. The theory of the Riematm Zeta-function (second edition revised by Heath-Brown, D. R.) (Clarendon Press, Oxford, 1986).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed