A note on multiplicative automatic sequences

Abstract

We prove that any q-automatic completely multiplicative function f : N → C essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming two standard conjectures in number theory, the methods allows for removing the assumption of completeness.

A note on multiplicative automatic sequences, II

“…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 .…”

On multiplicative automatic sequences

“…Relying on the following result, we can prove Proposition 3.1 by essentially the same methods which were used by Klurman and Kurlberg [8] for completely multiplicative sequences. Then there exists a Dirichlet character χ modulo M such that b(n) = χ(n) for all n ∈ N 0 coprime to M .…”

“…Relying on the following result, we can prove Proposition by essentially the same methods which were used by Klurman and Kurlberg for completely multiplicative sequences. Proposition There exists a threshold such that if is a prime, then for all .…”

“…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.…”