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Mahler series with multiplicative coefficient sequences

Jason Bell, Daniel Smertnig

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 Bézivin and Bell--Bruin--Coons) and of multiplicative automatic sequences (by Konieczny--Lemańczyk--Müllner). Both of these results are used in our characterization, so we do not obtain new proofs of these special cases.

Mahler series with multiplicative coefficient sequences

Abstract

We prove that every Mahler series, over a field of characteristic , 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 Bézivin and Bell--Bruin--Coons) and of multiplicative automatic sequences (by Konieczny--Lemańczyk--Müllner). Both of these results are used in our characterization, so we do not obtain new proofs of these special cases.
Paper Structure (16 sections, 39 theorems, 69 equations)

This paper contains 16 sections, 39 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Rational series and linear recurrence sequences
  5. Mahler series
  6. Regular sequences
  7. Automatic sequences
  8. Multiplicative k-regular sequences
  9. Mahler denominators
  10. Mahler denominators of series in K[[xl]]
  11. Mahler denominators under some algebraic operations
  12. Mahler denominators of rational functions
  13. Multiplicative Mahler series with non-completely multiplicative primes
  14. Rings of Mahler operators
  15. Mahler series with completely multiplicative primes.
  16. ...and 1 more sections

Key Result

theorem 1

Let $K$ be a field of characteristic $0$. Let $F = \sum_{n=0}^\infty f(n) x^n \in {K}\llbracket x \rrbracket$ be an algebraic series over $K(x)$. If $f\colon \mathbb N \to K$ is multiplicative, then there exist $r \ge 0$ and an eventually periodic multiplicative function $\chi\colon \mathbb N \to K$

Theorems & Definitions (81)

  • theorem 1: bell-bruin-coons12
  • theorem 2: konieczny-lemanczyk-muellner22
  • theorem 3
  • corollary 1
  • lemma 1
  • definition 1
  • definition 2
  • theorem 4: adamczewski-bell17
  • definition 3
  • definition 4
  • ...and 71 more