The sum of digits of Gaussian primes
type: Item
instance of: scholarly article
title: The sum of digits of Gaussian primes
zbMATH Open document ID: 1243.11007
author: Johannes F. Morgenbesser
published in: The Ramanujan Journal
publication date: 2012-02-13T00:00:00Z
review text: Let \(b\geq 2\) be an integer. \textit{C. Mauduit} and \textit{J. Rivat} [Ann. Math. (2) 171, No. 3, 1591--1646 (2010; Zbl 1213.11025)] settled a long-standing problem of Gelfond (1967/68) about the distribution of the sum of digits function \(s_b\) of primes in the digital representation of integers in base \(b\). They showed that \((s_b(p))\), \(p\) running through the primes, is uniformly distributed in arithmetic progressions, and that \((\alpha s_b(p))\) is uniformly distributed modulo 1 if and only if \(\alpha\) is irrational. One has a similar notion of the sum of digits function in the Gaussian integers, where the base \(b\) is replaced by \(q=-a+i\in\mathbb{Z}[i]\) with \(a\geq 1\). The Mauduit-Rivat result has been extended to the Gaussian primes by \textit{M. Drmota, J. Rivat} and the reviewer [Monatsh. Math. 155, No. 3-4, 317--347 (2008; Zbl 1190.11053)] but there were rather severe restrictions on the base \(q\) (\(q\) prime and sufficiently large in modulus). In the paper under review, the author obtains the complete result, and he puts it in an even more general setting. He shows that the sequence \((\alpha s_{q}(p))\) running through Gaussian primes that lie in a circular sector is uniformly distributed modulo 1 if and only if \(\alpha\) is irrational. From this, he also deduces well-distribution in arithmetic progressions. The proof involves, besides other tools, the use of a Vaughan-type identity, a suitable Gaussian integer version of the van der Corput inequality, discrete Fourier analysis in \(\mathbb{Z}[i]\) and a careful analysis of the carry propagation rule in this numeration system. The paper is very well written.
reviewed by: Thomas Stoll
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11L20
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 42A05
zbMATH DE Number: 6006054
zbMATH Keywords: Gaussian primes
zbMATH Keywords: Sum-of-digits function
zbMATH Keywords: Fourier analysis
zbMATH Keywords: Exponential sums
zbMATH Keywords: Beurling-Selberg function
MaRDI profile type: Publication
full work available at URL: s11139-010-9289-3
OpenAlex ID: W2051596298
cites work: The sum of digits of primes in \(\mathbb Z[i]\)
cites work: Sequences, discrepancies and applications
cites work: Sur les nombres qui ont des propriétés additives et multiplicatives données
cites work: Asymptotic normality of \(b\)-additive functions on polynomial sequences in the Gaussian number field
cites work: The Sum-of-Digits Function for Complex Bases
cites work: Q4830109
cites work: The sum of digits of squares
cites work: On a problem of Gelfond: the sum of digits of prime numbers
cites work: The sum of digits of squares in \(\mathbb Z[i]\)
cites work: Some extremal functions in Fourier analysis
DOI: 10.1007/S11139-010-9289-3
Recommended article: The sum of digits of primes in \(\mathbb Z[i]\)
Recommended article: The Sum-of-Digits Function for Complex Bases
Recommended article: The sum of digits of squares in \(\mathbb Z[i]\)
Recommended article: The distribution of generalized sum-of-digits functions in residue classes
Recommended article: Prime numbers with a positive proportion of preassigned digits
Recommended article: Additive functions with respect to expansions over the set of Gaussian integers
Recommended article: On Gelfond's conjecture about the sum of digits of prime numbers
Recommended article: On a problem of Gelfond: the sum of digits of prime numbers
Recommended article: g-ADDITIVE FUNCTIONS AND WELL DISTRIBUTION
Recommended article: Q3539407
description: scientific article; zbMATH DE number 6006054