Automatic sequences fulfill the Sarnak conjecture

TL;DR

This paper introduces a novel method for analyzing automatic sequences, enabling the proof of the Sarnak conjecture and a Prime Number Theorem for certain classes of automatic sequences, advancing understanding in number theory and automata.

Contribution

The paper develops a new approach to study automatic sequences, proving the Sarnak conjecture and establishing a Prime Number Theorem for specific automata-generated sequences.

Findings

Proves the Sarnak conjecture for automatic sequences.

Establishes a Prime Number Theorem for automata with strongly connected structure.

Introduces a new method to analyze automatic sequences.

Abstract

We present in this paper a new method to deal with automatic sequences. This method allows us to prove a M\"obius-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 $0$.