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Monatshefte für Mathematik

, Volume 187, Issue 4, pp 705–728 | Cite as

On the sum of digits of special sequences in finite fields

  • Cathy Swaenepoel
Article
  • 119 Downloads

Abstract

In \(\mathbb {F}_q\), Dartyge and Sárközy introduced the notion of digits and studied some properties of the sum of digits function. We will provide sharp estimates for the number of elements of special sequences of \(\mathbb {F}_q\) whose sum of digits is prescribed. Such special sequences of particular interest include the set of n-th powers for each \(n\ge 1\) and the set of elements of order d in \(\mathbb {F}_q^*\) for each divisor d of \(q-1\). We provide an optimal estimate for the number of squares whose sum of digits is prescribed. Our methods combine A. Weil bounds with character sums, Gaussian sums and exponential sums.

Keywords

Finite fields Digits Sum of digits Character sums Squares Primitive elements 

Mathematics Subject Classification

Primary 11A63 Secondary 11T23 11B83 

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Aix Marseille Université, CNRSCentrale Marseille, I2MMarseilleFrance
  2. 2.Institut de Mathématiques de Marseille UMR 7373 CNRSMarseille Cedex 9France

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