On multiplicative automatic sequences

Abstract

We show that any automatic multiplicative sequence either coincides with a Dirichlet character or is identically zero when restricted to integers not divisible by small primes. This answers a question of Bell, Bruin and Coons. A similar result was obtained independently by Klurman and Kurlberg.

Automatic sequences are orthogonal to aperiodic multiplicative functions

“…From now on, we consider only substitutions θ satisfying (15), (18), (25) and (26). Often, we will additionally assume that θ is pure.…”

“…That said, one realizes immediately that the answer to such a question is negative as periodic functions are automatic sequences and there are many examples of periodic, multiplicative functions. 3 Besides, even amongst non-periodic automatic sequences there are examples of (completely) multiplicative, zero mean functions [2], [40], [43], see also recent [24], [26] for a significant progress on characterizing such sequences. On the other hand, it has been proved in [13] that many automatic sequences given by so-called bijective substitutions are orthogonal to all zero mean, bounded, multiplicative functions (in [9], it is proved that they are orthogonal to the Möbius function).…”

“…Definition 5.1. We call Σ a substitution joining of θ and ζ if Σ satisfies (15) and (18) (note that (25) and (26) are satisfied automatically as θ, ζ are substitutions, so they satisfy these two properties).…”

“…Remark 8. However, if we replace H λ by H λ × Z/hZ, this assumption is satisfied and the proof of Proposition 7 remains the same 26. Note that we drop the assumption of ergodicity, cf.…”

Erratum

“…Between September 2019 and February 2020, twenty‐one research articles 1–21 were published in the Bulletin of the London Mathematical Society with incorrect copyright statements. The copyright in these articles is stated as belonging to the London Mathematical Society; however, that is incorrect and the copyright remains with the authors of those articles.…”