On the estimation of the second central moment for any additive arithmetic functions
type: Item
instance of: scholarly article
title: On the estimation of the second central moment for any additive arithmetic functions
zbMATH Open document ID: 0531.10054
DOI: 10.1007/BF00966361
published in: Lithuanian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \(f(m)\) be an additive arithmetic function, \[ A_ n(f)=\sum_{p\leq n}f(p)p^{-1},\quad D^ 2_ n(f)=\sum_{p^{\alpha}\leq n}| f^ 2(p^{\alpha})| \, p^{-\alpha}, \] \[ \tau_ n(f)=(D^ 2_ n(f)/n)\sum^{n}_{m=1}| f(m)-A_ n(f)|^ 2. \] \(\tau_ n\) be the supremum over all \(f\neq 0\) then \[ \tau_ n=1.5+O(\log^{-1/2}n). \] The same estimate is also obtained in the cases \(A_ n(f)=\sum_{p^{\alpha}\leq n}f(p^{\alpha})(1-p^{-1})p^{-\alpha}\) and \(A_ n(f)=\sum_{p^{\alpha}\leq n}f(p^{\alpha})p^{-\alpha}.\) Only estimates from above and below for \(\tau_ n\) were known earlier.
reviewed by: B. V. Levin
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N60
zbMATH DE Number: 3841993
zbMATH Keywords: second central moment
zbMATH Keywords: additive arithmetic functions
zbMATH Keywords: asymptotic estimate
MaRDI profile type: Publication
cites work: On the estimation of the second central moment for strongly additive arithmetic functions
cites work: Q3853725
full work available at URL: https://doi.org/10.1007/bf00966361
OpenAlex ID: W2030321436
author: Jonas Kubilius
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description: scientific article; zbMATH DE number 3841993