# On completely multiplicative automatic sequences

@article{Li2020OnCM,
title={On completely multiplicative automatic sequences},
author={Shuo Li},
journal={Journal of Number Theory},
year={2020}
}
• Shuo Li
• Published 2020
• Mathematics
• Journal of Number Theory
In this article we prove that all completely multiplicative automatic sequences $(a_n)_{n \in \mathbf{N}}$ defined on $\mathbf{C}$, vanishing or not, can be written in the form $a_n=b_n\chi_n$ where $(b_n)_{n \in \mathbf{N}}$ is an almost constant sequence, and $(\chi_n)_{n \in \mathbf{N}}$ is a Dirichlet character.
A note on multiplicative automatic sequences, II
• Mathematics
• 2019
We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L.Expand
How to prove that a sequence is not automatic
• Mathematics
• 2021
Automatic sequences have many properties that other sequences (in particular, non-uniformly morphic sequences) do not necessarily share. In this paper we survey a number of different methods that canExpand
On multiplicative automatic sequences
We obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.
Multiplicative automatic sequences
• Mathematics
• Mathematische Zeitschrift
• 2021
We obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

#### References

SHOWING 1-10 OF 17 REFERENCES
Rigidity theorems for multiplicative functions
• Mathematics
• 2017
We establish several results concerning the expected general phenomenon that, given a multiplicative function $$f:\mathbb {N}\rightarrow \mathbb {C}$$f:N→C, the values of f(n) and $$f(n+a)$$f(n+a)Expand
Subword complexity and non-automaticity of certain completely multiplicative functions
• Yining Hu
• Mathematics, Computer Science
• 2017
This article proves that for a completely multiplicative function f from $\mathbb{N}^*$ to a field $K$ such that the set p is finite, the asymptotic subword complexity of f is $\Theta(n^t)$, where $t$ is the number of primes p that f(p)neq 0_K, 1_K. Expand
Automatic sequences are orthogonal to aperiodic multiplicative functions
• Mathematics, Physics
• 2018
Given a finite alphabet $\mathbb{A}$ and a primitive substitution $\theta:\mathbb{A}\to\mathbb{A}^\lambda$ (of constant length $\lambda$), let $(X_\theta,S)$ denote the corresponding dynamicalExpand
An averaged Chowla and Elliott conjecture along independent polynomials
We generalize a result of Matomaki, Radziwill, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or moreExpand
Furstenberg Systems of Bounded Multiplicative Functions and Applications
• Mathematics
• International Mathematics Research Notices
• 2019
We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that theseExpand
Automatic Dirichlet Series
• Mathematics
• 2000
Abstract Dirichlet series whose coefficients are generated by finite automata define meromorphic functions on the whole complex plane. As consequences, a new proof of Cobham's theorem on theExpand
Mock characters and the Kronecker symbol
• Mathematics
• 2016
We introduce and study a family of functions we call the "mock characters". These functions satisfy a number of interesting properties, and of all completely multiplicative arithmetic functions seemExpand
Completely Multiplicative Automatic Functions
Abstract We show that a completely multiplicative automatic function which does not have 0 as a value is almost periodic.
Uniform tag sequences
• A. Cobham
• Mathematics, Computer Science
• Mathematical systems theory
• 2005
Structural properties, local and asymptotic, of members of a class of simple, realtime generable sequences are analyzed.
Mathematische Annalen
Nutzungsbedingungen DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich fürExpand