Mahler series with multiplicative coefficient sequences

TL;DR

This paper proves that Mahler series with multiplicative coefficients over characteristic zero fields are regular and provides an explicit characterization, extending previous results on rational and automatic sequences.

Contribution

It offers a new explicit characterization of Mahler series with multiplicative coefficients, unifying and extending prior classifications of rational and automatic sequences.

Findings

All such Mahler series are regular.

Explicit characterization of these series.

Extension of known classifications of rational and automatic sequences.

Abstract

We prove that every Mahler series, over a field of characteristic $0$, with multiplicative coefficients is regular in the sense of Allouche and Shallit. We also obtain an explicit characterization of such series. This yields a joint extension of the characterization of rational series with multiplicative coefficients (by B\'ezivin and Bell--Bruin--Coons) and of multiplicative automatic sequences (by Konieczny--Lema\'nczyk--M\"ullner). Both of these results are used in our characterization, so we do not obtain new proofs of these special cases.