Bounds for the local distance between arithmetical distributions
type: Item
instance of: scholarly article
title: Bounds for the local distance between arithmetical distributions
zbMATH Open document ID: 1194.11082
author: Jonas Šiaulys
author: A. Machyulis
published in: Acta Applicandae Mathematicae
publication date: 2007-07-19T00:00:00Z
review text: Let \(f_x\) be a set of real-valued strongly additive functions, and let \(\xi_{xp},\, p\leq x\) be independent random variables (for every fixed \(x\)), defined by \[ \xi_{xp} = f_x(p) \text{ with probability \(\frac1p\) }, \text{ and } = 0 \text{ with probability \(1 - \frac1p\) }, \] and define the distribution function \[ P_x(u) = P\left( \sum_{p\leq x} \xi_{xp} < u \right). \] Using (for some condition \(A_x\)) the abbreviations \[ \nu_x(A_x) = \frac1{[x]} \cdot \#\{ n\leq x;\; n\in A_x\},\qquad \nu_x= \nu_x(f_x(n) < u), \] the authors define, for integer-valued additive functions \(f_x\), a ``local distance'' \[ \rho(\nu_x, P_x) = \sup_{k\in\mathbb Z} \left| \nu_x(f_x(n)=k) - P\left( \sum_{p\leq x} \xi_{xp} = k \right) \right|. \] In [Lith. Math. J. 45, No. 4, 487--492 (2005); translation from Liet. Mat. Rink. 45, No. 4, 603--610 (2005; Zbl 1194.11083)], the authors gave an upper estimate for this local distance. Now, the authors prove a lower estimate under the additional assumption, that all the values \(f_x(p)\) are integers in \(\{0,1,2,\dots,C\}\): For some absolute constant \(c_1\) \[ \exp\left\{ - \frac{37}{\gamma^2(x)} \, \log \frac{3C}{r(x)} \right\} \cdot \mathbf 1_{\{\gamma(x) \geq 7.5 \frac{\log\log x}{\log x}\}} \leq \rho(\nu_x,\, P_x) \leq c_1 \left\{ \sqrt{r(x)\,} + x^{-\frac45} \right\}. \] Here \(\mathbf 1_{\dots}\) is the characteristic function of the indicated set, and \[ \gamma(x) = \frac1{\log x} \; \sum_{{p\,\leq x}\atop {f_x(p)\not=0}} \frac{\log p}p, \quad E(x) = \sum_{{p\,\leq x}\atop {f_x(p)\not=0}} \frac1p, \quad r(x) = \min\left( \gamma(x),\, \frac1{E(x)} \right). \]
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 5172640
zbMATH Keywords: integer-valued, strongly additive functions
zbMATH Keywords: local distance
zbMATH Keywords: distribution functions of additive functions
zbMATH Keywords: Lévy distance
zbMATH Keywords: convergence of the local distance to zero
reviewed by: Wolfgang Schwarz
MaRDI profile type: Publication
full work available at URL: s10440-007-9123-0
OpenAlex ID: W2033274190
cites work: Q5343492
cites work: Q3867876
cites work: Q3867877
cites work: Q4252249
cites work: On the distribution of additive arithmetic functions
cites work: On the local distance between arithmetical distributions
cites work: Kubilius-type sequences of additive functions
DOI: 10.1007/S10440-007-9123-0
description: scientific article; zbMATH DE number 5172640