Joint distribution of completely \(q\)-additive functions in residue classes
type: Item
instance of: scholarly article
title: Joint distribution of completely \(q\)-additive functions in residue classes
zbMATH Open document ID: 1335.11079
author: Daniel Berend
author: G. A. Kolesnik
published in: Journal of Number Theory
publication date: 2015-12-07T00:00:00Z
review text: An arithmetic function \(f\) is said to be completely \(q\)-additive, if \(f(0)=0\), and \(f(n)=\sum_{i\geq 0}f(a_i)\) for \(n=\sum_{i\geq0}a_iq^i\). The authors study the joint distribution of completely \(q_j\)-additive functions in residue classes. Namely, if \(f_i\) is \(q_i\)-additive (\(i=1,\dots,\ell\)) with pairwise primes \(q_i\), then the cardinality of the set \[ \{1\leq n\leq N\mid f_j(a_jn+b_j)\equiv c_j\pmod{m_j},\,1\leq j\leq\ell\} \] is asymptotically equal to \(cN+O(N^{1-\delta})\) (if it is not zero). Here, \(c\) and \(\delta\) are explicitly given. Similar result holds in short intervals, i.e., when \(M\leq n\leq N+M\). The above theorems are significant generalizations of the results of \textit{D.-H. Kim} [J. Number Theory 74, No. 2, 307--336 (1999; Zbl 0920.11067)].
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 11N69
zbMATH DE Number: 6516893
zbMATH Keywords: uniform distribution
zbMATH Keywords: joint distribution
zbMATH Keywords: \(q\)-additive function
zbMATH Keywords: completely \(q\)-additive function
reviewed by: István Mező
MaRDI profile type: Publication
full work available at URL: j.jnt.2015.09.006
OpenAlex ID: W1904304252
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DOI: 10.1016/J.JNT.2015.09.006
description: scientific article; zbMATH DE number 6516893