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Primes with an average sum of digits

Part of: Exponential sums and character sums Elementary number theory Multiplicative number theory Limit theorems

Published online by Cambridge University Press:  01 March 2009

Michael Drmota ,
Christian Mauduit  and
Joël Rivat
Michael Drmota
Affiliation:
Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, A-1040 Wien, Austria (email: michael.drmota@tuwien.ac.at)
Christian Mauduit
Affiliation:
Institut de Mathématiques de Luminy, CNRS UMR 6206, Université de la Méditerranée, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: mauduit@iml.univ-mrs.fr)
Joël Rivat
Affiliation:
Institut de Mathématiques de Luminy, CNRS UMR 6206, Université de la Méditerranée, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: rivat@iml.univ-mrs.fr)
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Abstract

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The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form $\sum _{p\le x} e(\alpha s_q(p))$ (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

Keywords

sum-of-digits functionprimesexponential sumscentral limit theorem

MSC classification

Secondary: 11A63: Radix representation; digital problems 11L03: Trigonometric and exponential sums, general 11N05: Distribution of primes 11L20: Sums over primes 60F05: Central limit and other weak theorems

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 271 - 292
Copyright
Copyright © Foundation Compositio Mathematica 2009

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