Rate of divergence of some integrals
type: Item
instance of: scholarly article
title: Rate of divergence of some integrals
zbMATH Open document ID: 0855.11051
DOI: 10.1007/BF02304106
author: A. Yu Popov
author: Sergei Konyagin
published in: Mathematical Notes
publication date: 1996-12-03T00:00:00Z
review text: Let \[ M(x)= \sum_{n\leq x} \mu (n), \qquad \Psi (x)= \sum_{n\leq x} \Lambda (n), \qquad B(x)= \sum_{mn\leq x} {\textstyle {1\over m}} \mu(m), \] where \(\mu (n)\) is the Möbius function and \(\Lambda (n)\) the Mangoldt function. The authors state a general Tauberian theorem and derive from it \[ \int^x_1 (\Psi (t) -t)^2 t^{-2} dt\geq c_1\log x, \quad \int^x_1 M^2 (t) t^{-2} dt\geq c_2 \log x, \quad \int^x_1 B^2 (t) t^{-2} dt\geq c_3 \log x. \]
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 40E05
Mathematics Subject Classification ID: 11M45
Mathematics Subject Classification ID: 40A10
zbMATH DE Number: 898926
zbMATH Keywords: asymptotic results
zbMATH Keywords: Möbius function
zbMATH Keywords: Mangoldt function
zbMATH Keywords: Tauberian theorem
reviewed by: Ekkehard Krätzel
MaRDI profile type: Publication
cites work: Q5523060
cites work: Q4071794
cites work: Q3735790
cites work: Q3783344
full work available at URL: https://doi.org/10.1007/bf02304106
OpenAlex ID: W1969404962
description: scientific article; zbMATH DE number 898926