About a theorem of Paolo Codecà's and \(\Omega\)-estimates for arithmetical convolutions: Addendum
type: Item
instance of: scholarly article
title: About a theorem of Paolo Codecà's and \(\Omega\)-estimates for arithmetical convolutions: Addendum
zbMATH Open document ID: 0732.11048
DOI: 10.1016/0022-314X(90)90092-6
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: The condition of the author's Theorem 1 in [J. Number Theory 30, 71-85 (1988; Zbl 0649.10034)] that the parameter \(B\geq 0\) has to be an integer is now removed. This version of the theorem can be used to obtain two- sided \(\Omega\)-estimates for the class of convolutions \[ g(x):=\sum_{n\leq x^{3/4}}\alpha (n)n^ af(x/n), \] where a,x\(\in {\mathbb{R}}:\) \(a\leq -1\), \(x>0\), \(\alpha (n)=1\) or \(\alpha (n)=\mu (n)\) (Möbius function) and f() is a 1-periodic function satisfying a Kubert identity of order \(1\leq \ell \in {\mathbb{N}}:\) \[ f(my)=m^{\ell - 1}\sum^{m-1}_{n=0}f(y+n/m)\quad \forall y\in {\mathbb{R}},\quad m\in {\mathbb{N}}. \] As an application, the author proves new or improved \(\Omega_{\pm}\)-results for the divisor functions \(G_{a,k}(x)\) of Chowla and Walum, the P-function of Hardy and Littlewood and to an error term related to the Euler \(\phi\)-function. Moreover, the constants implied by the \(\Omega\)-symbol can be computed.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N56
zbMATH DE Number: 4210264
zbMATH Keywords: divisor functions of Chowla and Walum
zbMATH Keywords: P-function of Hardy and Littlewood
zbMATH Keywords: Euler \(\phi \) -function
author: Y.-F. S. Pétermann
reviewed by: Winfried Recknagel
MaRDI profile type: Publication
cites work: On the Changes of Sign of a Certain Error Function
cites work: Notes on the Theory of Series (XX): On Lambert Series
cites work: An extremal result for divisor functions
cites work: Fluctuations in the mean of Euler's phi function
cites work: About a theorem of Paolo Codecà's and \(\Omega\)-estimates for arithmetical convolutions
cites work: An \(\Omega\)-theorem \dots : Addendum
cites work: Q5666618
cites work: Mertens' theorem for arithmetic progressions
full work available at URL: 0022-314x(90)90092-6
OpenAlex ID: W2006880538
description: scientific article; zbMATH DE number 4210264