Automatic sequences fulfill the Sarnak conjecture

Abstract

We present in this paper a new method to deal 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 $0$.

55 pages

References in corpus (2)

• Möbius disjointness for homogeneous dynamics
• Substitutions and Möbius disjointness

Cited by in corpus (10)

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• Automorphisms of automatic shifts
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• Sarnak's Conjecture for nilsequences on arbitrary number fields and applications
• Chowla and Sarnak Conjectures for Kloosterman Sums
• Möbius disjointness for a class of exponential functions
• Gowers norms for automatic sequences
• Bracket words along Hardy field sequences
• The Thue-Morse and Rudin-Shapiro sequences at primes in principal number fields
• On subwords in the base-$q$ expansion of polynomial and exponential functions