On completely multiplicative automatic sequences

Abstract

In this article, we prove that all completely multiplicative automatic sequences (an)n∈N defined on C, vanishing or not, can be written in the form an = bnχn, where (bn)n∈N is an almost constant sequence, and (χn)n∈N is a Dirichlet character.

A note on multiplicative automatic sequences, II

“…Allouche and Goldmakher considered a related question classifying what they called ‘mock’ Dirichlet characters. Finally Li and the authors very recently proved Conjecture when is additionally assumed to be completely multiplicative. Moreover, the methods of settled Conjecture fully under the assumption of the Generalized Riemann Hypothesis.…”

“…A function is called multiplicative if for all pairs . The question of which multiplicative functions are ‐automatic attracted considerable attention of several authors including . In particular, the following conjecture was made in .…”

Multiplicative automatic sequences

“…This has been strengthened in [20] to Weyl almost periodicity (in the primitive case). Relatively recently, multiplicative automatic sequence have been studied in numerous papers, [1,3,4,6,7,12,[17][18][19]21,22,26], and [30]. A particular focus was put on the study of the case of completely multiplicative automatic sequences [1,21] with a complete classification given in [22].…”

On multiplicative automatic sequences

“…As such, for any class of sequences it is natural to inquire into which sequences in it are automatic. In particular, the question of classifying automatic multiplicative sequences has been investigated by a number of authors, including [1,2,7,8,12,14]. The interplay between multiplicative and automatic sequences is studied also in [3,5,11,13,15], among others.…”

“…The two most recent papers listed above give a classification of completely multiplicative automatic sequences, but until now the question remained open for sequences which are multiplicative but not completely so. In particular, the authors of conjectured that a multiplicative automatic sequence agrees with an eventually periodic sequence on the primes.…”