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Monatshefte für Mathematik

, Volume 155, Issue 3–4, pp 317–347 | Cite as

The sum of digits of primes in \({\mathbb{Z}}\)[i]

  • Michael DrmotaEmail author
  • Joël Rivat
  • Thomas Stoll
Article

Abstract

We study the distribution of the complex sum-of-digits function s q with basis q = –a±i, \({a \in \mathbb{Z}^+}\) for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat (http://iml.univ-mrs.fr/~rivat/publications.html) for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αs q (p)) provided \({\alpha \in \mathbb{R} \setminus \mathbb{Q}}\) and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m.

Keywords

Sum-of-digits function Primes Gaussian integers Exponential sums 

Mathematics Subject Classification (2000)

Primary: 11A63 Secondary: 11N60 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryTechnische Universität WienWienAustria
  2. 2.Institut de Mathématiques de Luminy, CNRS UMR 6206Université de la MéditerranéeMarseille Cedex 9France

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