Mahler series with multiplicative coefficient 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.