Distribution of complex algebraic numbers
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- by Friedrich Götze, Dzianis Kaliada and Dmitry Zaporozhets PDF
- Proc. Amer. Math. Soc. 145 (2017), 61-71 Request permission
Abstract:
For a region $\Omega \subset \mathbb {C}$ denote by $\Psi (Q;\Omega )$ the number of complex algebraic numbers in $\Omega$ of degree $\leq n$ and naive height $\leq Q$. We show that \[ \Psi (Q;\Omega )=\frac {Q^{n+1}}{2\zeta (n+1)}\int _\Omega \psi (z) \nu (dz)+O\left (Q^n \right ),\quad Q\to \infty , \] where $\nu$ is the Lebesgue measure on the complex plane and the function $\psi$ will be given explicitly.References
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Additional Information
- Friedrich Götze
- Affiliation: Department of Mathematics, Bielefeld University, P.O. Box 10 01 31, 33501 Bielefeld, Germany
- MR Author ID: 194198
- Email: goetze@math.uni-bielefeld.de
- Dzianis Kaliada
- Affiliation: Institute of Mathematics, National Academy of Sciences of Belarus, Surganova Str 11, 220072 Minsk, Belarus
- MR Author ID: 933471
- Email: koledad@rambler.ru
- Dmitry Zaporozhets
- Affiliation: St. Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, 191011 St. Petersburg, Russia
- MR Author ID: 744268
- Email: zap1979@gmail.com
- Received by editor(s): October 16, 2015
- Received by editor(s) in revised form: March 17, 2016
- Published electronically: July 7, 2016
- Additional Notes: This research was supported by CRC 701, Bielefeld University (Germany).
- Communicated by: Matthew A. Papanikolas
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 61-71
- MSC (2010): Primary 11N45; Secondary 11C08
- DOI: https://doi.org/10.1090/proc/13208
- MathSciNet review: 3565360