On the \(B^{\lambda}\) almost periodic behaviour of certain arithmetical convolutions
type: Item
instance of: scholarly article
title: On the \(B^{\lambda}\) almost periodic behaviour of certain arithmetical convolutions
zbMATH Open document ID: 0552.10030
author: P. Codecà
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1984-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/107964
review text: The author is interested in the functions \[ P(x)=\sum_{n\leq x}(1/n)\cos (x/n), \quad Q(x)=\sum_{n\leq x}(1/n)\sin (x/n), \] \[ R_1(x)=-\sum_{n\leq x}(1/n)(\{x/n\}-1/2),\quad R_2(x)=-\sum_{n\leq x}(\mu (n)/n)(\{x/n\}- 1/2); \] the latter two are connected with the asymptotic formulas for \(\sum_{n\leq x}\sigma (n)/n\), resp. \(\sum_{n\leq x}\phi (n)/n\). All these functions are unbounded; however, as a corollary to his main theorems, the author shows that \(E(x)=E_1(x)+O(1),\) where \(E_1\) is \(B^{\lambda}\)-almost-periodic, and that \((1/x)\int^{x}_{1}| E(t)|^{\lambda} \, dt=O(1)\) for any \(\lambda >0\), where \(E(x)\) is one of the four functions \(P(x),\ldots,R_2(x)\). The author's main result is as follows: Define the convolution \[ g(x)=\sum_{n\leq y(x)}(a(n)/n)f(x/n), \] where \(a(n)\) is a bounded sequence of real numbers, \(y(x)\nearrow \infty\), as \(x\to \infty\), but \(y(x)x^{-\varepsilon}\to 0\) for any \(\varepsilon >0\), and where \(f\) is a real-valued 1-periodic function of bounded variation satisfying \(\int^{1}_{0}f(x) \,dx=0\). Then \(g\) is \(B^{\lambda}\)-almost-periodic for any \(\lambda >0\), and the mean value \[ M(g^k)=\lim_{x\to \infty}(1/x)\int^{x}_{1}g(t)^k \,dt \] exists for any integer \(k>0\).
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 42A75
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 42A85
zbMATH DE Number: 3879029
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: \(B^{\lambda }\)-almost-periodic
zbMATH Keywords: convolution
zbMATH Keywords: function of bounded variation
zbMATH Keywords: mean value
reviewed by: Wolfgang Schwarz
MaRDI profile type: Publication
cites work: Q5338374
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description: scientific article; zbMATH DE number 3879029