Sur certaines équations fonctionnelles arithmétiques. (On certain arithmetical functional equations)
type: Item
instance of: scholarly article
title: Sur certaines équations fonctionnelles arithmétiques. (On certain arithmetical functional equations)
zbMATH Open document ID: 0968.11033
author: Régis de la Bretèche
author: Gérald Tenenbaum
publication date: 2000-11-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75461
review text: The motivation for this extensive study of certain classes of additive and multiplicative arithmetical functions is the additive function, called by the authors the Gutman-Ivić-Matula function [see \textit{I. Gutman} and the reviewer, Discrete Math. 150, 131-142 (1996; Zbl 0854.05035)]. It is the unique completely additive function satisfying \(f(p_k) = 1 + f(k),\) where \(p_k\) is the \(k\)-th prime. The authors consider more generally \(\mathcal E\), the class of completely additive functions which satisfy \[ F(p_k) = F(k) + b\log\log k + a + O\left({\log\log k\over\log k}\right) \] for some \(a,b\in\mathbb R\). Théorème 1.1 evaluates the mean value of any arithmetical function \(g(n)\) satisfying \(g(n) \ll \log^\alpha n (\alpha > 0)\) in terms of \[ R(x,g) := {1\over x}\sum_{n\leq x}g(n) - {1\over x}\sum_{p_k\leq x}g(k)\left[{x\over p_k}\right]. \] The method of proof is based on the use of certain approximate functional equations of independent interest. In Théorème 1.2 they obtain the asymptotic evaluation of moments of functions from \(\mathcal E\), more precisely for \(f_0 := f - \Phi_1(f)\log\), where \(f\in {\mathcal E}\) and \(\Phi_1(f)\) is a suitable constant depending on \(f\). From this result they deduce the following one. Let \(f\) be a completely additive arithmetical non-zero function satisfying \(f(p_k) \geq f(k)\) for \(k\in \mathbb N\). Then \[ \sum_{n\leq x}f(n)\gg x\log x, \] which implies that the constant \(\Phi_1(f)\) is strictly positive. Also if \(f\in {\mathcal E}\) and the second moment of \(f_0\) (defined above) is \(o(x\log x)\) as \(x\to\infty\), then \(f(n) = \Phi_1(f)\log n\). The remaining two theorems of the paper deal with asymptotic results on certain multiplicative functions of at most unit modulus. They are used to yield information on the distribution of values of functions from \({\mathcal E}\). For example, if \[ h_N(n;\tau) = \exp(i\tau f_0(n)/\sqrt{\Phi_2\log N}) \quad(\tau\in\mathbb R, N\geq 2), \] where \(\Phi_2\) is …
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 05C90
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 11N64
zbMATH DE Number: 1528712
zbMATH Keywords: additive arithmetical functions
zbMATH Keywords: multiplicative arithmetical functions
zbMATH Keywords: Gutman-Ivić-Matula function
zbMATH Keywords: distribution of prime numbes
zbMATH Keywords: Berry-Esseen inequality
zbMATH Keywords: Fourier-Stieltjes transform
zbMATH Keywords: characteristic function
MaRDI profile type: Publication
OpenAlex ID: W2079120280
cites work: Q3219268
cites work: Function spaces and continuous algebraic pairings for varieties
cites work: Q4313516
cites work: On Matula numbers
cites work: Q4296347
cites work: Q3123961
DOI: 10.5802/AIF.1797
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1528712