Natural divisors and the Brownian motion
type: Item
instance of: scholarly article
title: Natural divisors and the Brownian motion
zbMATH Open document ID: 0864.11040
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-11-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93646
full work available at URL: jtnb8-1.html
review text: On the space of integers \(1\leq m\leq x\), each having probability \(1/x\), we consider the random variables \[ X_x(m,t) =(\log \log x)^{-1/2} \biggl(\log_2 \tau\bigl(m, \exp (\log x)^t-t \log \log x\bigr) \biggr) \] as a random process, with \(t\) changing from 0 to 1. Here \(\tau (m,u)\) is the number of divisors of \(m\) up to \(u\). This process is shown to converge (as \(x\to\infty)\) to the Wiener process. This continues the author's investigations started in [Acta Math. Hung. 67, 1-17 (1995; Zbl 0823.11045)]. A version of the arcsine law is also proved. An arcsine law for the divisors was obtained by \textit{J.-M. Deshouillers}, \textit{F. Dress} and \textit{G. Tenenbaum} [Acta Arith. 34, 273-285 (1979; Zbl 0408.10035)], but this one is based on a different model and is essentially analogous to the arcsine law of \textit{P. Erdös} [Aequationes Math. 2, 177-183 (1969; Zbl 0174.08104)] for prime divisors.
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 912504
zbMATH Keywords: arithmetical functions
zbMATH Keywords: Brownian motion
zbMATH Keywords: random variables
zbMATH Keywords: random process
zbMATH Keywords: Wiener process
zbMATH Keywords: arcsine laws
zbMATH Keywords: divisors
author: Eugenijus Manstavičius
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
OpenAlex ID: W2330776198
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DOI: 10.5802/JTNB.162
description: scientific article; zbMATH DE number 912504