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THE MINIMAL GROWTH OF A $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-REGULAR SEQUENCE

Part of: Sequences and sets Multiplicative number theory

Published online by Cambridge University Press:  22 May 2014

JASON P. BELL ,
MICHAEL COONS  and
KEVIN G. HARE
JASON P. BELL
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Canada email jpbell@uwaterloo.ca
MICHAEL COONS*
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia email Michael.Coons@newcastle.edu.au
KEVIN G. HARE
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Canada email kghare@uwaterloo.ca
*
✉Michael.Coons@newcastle.edu.au
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Abstract

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We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to \mathbb{Z}$ is an unbounded $k$-regular sequence, we show that there is a constant $c>0$ such that $|f(n)|>c\log n$ infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.

Keywords

automata sequencesregular sequencesgrowth of arithmetic functions

MSC classification

Secondary: 11B85: Automata sequences 11N56: Rate of growth of arithmetic functions 11B37: Recurrences
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 195 - 203
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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