On a conjecture of Kátai for additive functions
type: Item
instance of: scholarly article
title: On a conjecture of Kátai for additive functions
zbMATH Open document ID: 0845.11030
author: Eduard Wirsing
author: Yuansheng Tang
author: Pintsung Shao
published in: Journal of Number Theory
publication date: 1996-09-11T00:00:00Z
review text: Let \(|x|\) denote the distance of \(x\) from the nearest integer and \(\Delta\) the forward difference of an arithmetic function \(f: \Delta f(n)= f(n+ 1)- f(n)\). The authors give two proofs of Kátai's conjecture: if \(f: \mathbb{N}\to \mathbb{R}\) is additive and \((*)\) \(|\Delta f(n)|\to 0\), then there exists \(\tau\in \mathbb{R}\) such that \[ |f(n)- \tau\log n|= 0\quad \text{for all } n. \] Weaker versions (with stronger conditions than \((*)\)) are already known.
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N64
zbMATH DE Number: 874189
zbMATH Keywords: additive functions
zbMATH Keywords: forward difference
zbMATH Keywords: arithmetic function
zbMATH Keywords: Kátai's conjecture
reviewed by: Jürgen Spilker
MaRDI profile type: Publication
Wikidata QID: Q123299328
full work available at URL: jnth.1996.0027
OpenAlex ID: W1965718267
DOI: 10.1006/JNTH.1996.0027
description: scientific article; zbMATH DE number 874189