Duke Mathematical Journal

Automatic sequences fulfill the Sarnak conjecture

Clemens Müllner

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Abstract

We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Möbius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to 0.

Article information

Source
Duke Math. J., Volume 166, Number 17 (2017), 3219-3290.

Dates
Received: 4 May 2016
Revised: 21 April 2017
First available in Project Euclid: 5 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1507169019

Digital Object Identifier
doi:10.1215/00127094-2017-0024

Mathematical Reviews number (MathSciNet)
MR3724218

Zentralblatt MATH identifier
06825580

Subjects
Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16}
Secondary: 11B85: Automata sequences 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 11N05: Distribution of primes 11L20: Sums over primes

Keywords
automata sequence Sarnak conjecture symbolic dynamics sums over primes exponential sums

Citation

Müllner, Clemens. Automatic sequences fulfill the Sarnak conjecture. Duke Math. J. 166 (2017), no. 17, 3219--3290. doi:10.1215/00127094-2017-0024. https://projecteuclid.org/euclid.dmj/1507169019


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