A new characteristic of the identity function
type: Item
instance of: scholarly article
title: A new characteristic of the identity function
zbMATH Open document ID: 0871.11063
author: Jean-Marie De Koninck
author: Imre Kátai
author: Bui Minh Phong
published in: Journal of Number Theory
publication date: 1997-10-01T00:00:00Z
review text: The following Theorem is proved. Let \(f: \mathbb{N}\to \mathbb{C}\) be a multiplicative function such that \(f(1)=1\) and \(f(p+m^2)=f(p)+f(m^2)\) for all primes \(p\) and integers \(m\geq 1\). Then \(f(n)=n\) for all integers \(n\geq 1\). The elementary but ingenious proof uses sieve results and explicit numerical bounds for various prime number functions. The result should be compared with that of \textit{C. Spiro} [J. Number Theory 42, 232-246 (1992; Zbl 0756.11027)], where the same conclusion is drawn from the assumption \(f(p+q)=f(p)+f(q)\) (\(p\), \(q\) prime).
Mathematics Subject Classification ID: 11N64
zbMATH DE Number: 1011057
zbMATH Keywords: bounds for prime number functions
zbMATH Keywords: multiplicative function
reviewed by: Dieter Wolke
MaRDI profile type: Publication
full work available at URL: jnth.1997.2108
OpenAlex ID: W1993240810
cites work: Q5542212
cites work: Q4052222
cites work: Q3259107
cites work: Explicit Estimates for θ (x; 3, l) and ψ(x; 3, l)
cites work: Approximate formulas for some functions of prime numbers
cites work: Additive uniqueness sets of arithmetic functions
DOI: 10.1006/JNTH.1997.2108
Recommended article: Q4387006
Recommended article: Q3106159
Recommended article: Q4643144
Recommended article: Q5016598
Recommended article: A Function-Theoretic Identity
Recommended article: Q3428773
Recommended article: Q5471944
Recommended article: A functional identity characterizing polynomials
Recommended article: A characterization of the identity with functional equations. II
Recommended article: Q3319185
description: scientific article; zbMATH DE number 1011057