The sum of digits of squares in \(\mathbb Z[i]\)
type: Item
instance of: scholarly article
title: The sum of digits of squares in \(\mathbb Z[i]\)
zbMATH Open document ID: 1203.11054
author: Johannes F. Morgenbesser
published in: Journal of Number Theory
publication date: 2010-07-20T00:00:00Z
review text: Let \(q\) and \(n\) be natural numbers and denote by \(s_q(n)\) the sum of digits of \(n\) written in base \(q\). Answering a question posed by \textit{A. O. Gelfond, C. Mauduit} and \textit{J. Rivat} [''La somme des chiffres des carrés,'' Acta Math. 203, No. 1, 107--148 (2009; Zbl 1278.11076)] showed that the sum \(s_q(n^2)\) of digits of squares is equidistributed in arithmetic progressions and the sequence \((\alpha s_q(n^2))_{n\in \mathbb{N}}\) is uniformly distributed modulo 1 if \(\alpha\) is irrational. This result hinges on estimates for exponential sums of the form \[ \sum\limits_{n\leq N} e(\alpha s_q(n^2)). \] The paper under review studies the analog of this question for the ring of Gaussian integers \(\mathbb{Z}[i]\). An estimate is established for exponential sums of the form \[ \sum\limits_{z\in \mathcal{D}_N} e(\alpha s_q(z^2)), \] where \(s_q(x)\) is an analog of the sum of digits function for \(\mathbb{Z}[i]\) and \((\mathcal{D}_N)_{N\in \mathbb{N}}\) is a \(\kappa\)-\(\mathbb{Z}[i]\)-sequence of subsets of \(\mathbb{Z}[i]\) with \(0<\kappa\leq 1/2\) (we refrain from giving the definition of such sequences here). Using this, the author deduces an equidistribution result on the sequence \(s_q(z^2)\) in arithmetic progressions and shows that \((\alpha s_q(z^2))_{z\in \mathbb{Z}[i]}\) is uniformly distributed modulo 1 if \(\alpha\) is irrational. The method is based on that of Mauduit and Rivat for the real case. However, a great deal of work is required to make this method working for \(\mathbb{Z}[i]\). In particular, the author needs to prove a suitable van der Corput type bound in this context.
reviewed by: Stephan Baier
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11N64
zbMATH DE Number: 5757993
zbMATH Keywords: sum-of-digits function
zbMATH Keywords: Gaussian integers
zbMATH Keywords: exponential sums
zbMATH Keywords: Følner sequences
zbMATH Keywords: squares
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.02.011
OpenAlex ID: W2036080649
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DOI: 10.1016/J.JNT.2010.02.011
description: scientific article; zbMATH DE number 5757993