On the sum of digits functions for number systems with negative bases
type: Item
instance of: scholarly article
title: On the sum of digits functions for number systems with negative bases
zbMATH Open document ID: 0987.11007
DOI: 10.1023/A:1009831204394
author: Peter J. Grabner
author: Jörg M. Thuswaldner
published in: The Ramanujan Journal
publication date: 2002-04-07T00:00:00Z
review text: Let \(q\geq 2\) be a fixed integer. Any integer \(n\) can be represented uniquely in the negative base \(-q\), i.e. in the form \(n=c_0+c_1(-q)+\cdots +c_h(-q)^h\), where \(c_i\in \{0,1,\dots ,q-1\}\). The authors investigate the sum of digits function \(\nu_{- q}(n)=c_0+c_1+\cdots +c_h\). They show, among other things, that \[ \sum_{n<N} (\nu_{-q}(n)-\nu_{-q}(-n)) =NG(\log_{q^2} N) +O(\log^2 N), \] where \(G(x)\) is a continuous, periodic, nowhere differentiable function, and obtain a Gaussian asymptotic distribution result for \(\nu_{-q}(n)-\nu_{-q}(-n)\). The proofs use automata and analytic methods.
reviewed by: László Tóth
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 1546949
zbMATH Keywords: digital expansion
zbMATH Keywords: sum of digits
zbMATH Keywords: finite automata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1546949