On a method of estimating from below the number of sign changes of arithmetic functions.
type: Item
instance of: scholarly article
title: On a method of estimating from below the number of sign changes of arithmetic functions.
zbMATH Open document ID: 0766.11037
DOI: 10.1007/BF01196856
author: B. Szydło
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: Let \(V(\Delta,X)\) denote the number of sign-changes in \((0,X]\), \((X>0)\), of the remainder term \(\Delta(x)=\psi(x)-x\) in the prime number theorem, where \(\psi(x)=\sum_{p^ r\leq x} \log p\) stands for the Chebyshev's function. Assume that \((A_ H)\) for all non-trivial zeros \(\rho=\beta+i\gamma\) of the Riemann zeta function \(\zeta\) with \(|\gamma|<H\) we have \(\beta=1/2\). Then the earlier author's result [Math. Ann. 283, 151-163 (1989; Zbl 0636.10035); see also ibid. 283, 139-149 (1989; Zbl 0636.10036)] is basically as follows: \[ V(\Delta,X)\geq(1-b/H)(\gamma_ 0/\pi)\log X \qquad (\log X\geq BH^ c) \tag{1} \] with \(b=3\) and \(B=0.09\), if \(H\geq 4400\), where \(\gamma_ 0=14.13\dots\) is the imaginary part of the first non-trivial zero of \(\zeta\). It is now natural to ask about efficiency of the method used by the author in the proof of (1). More specifically, one may ask about the best constants \(b\) and \(B\) in (1) obtainable when using the method. The author succeeds in answering this question and gives the lower limits for \(b\) and \(B\). In particular, he remarks that there is an explicitly given constant \(b_ 0>0\) \(b_ 0=1.242\dots\)) so that the inequality (1) with \(b<b_ 0\) is not attainable by means of the present method under the assumption \((A_ H)\) for any \(H\). It is to mention that if one limits oneself to the basic problem of determination of \(\ell=\liminf_{X\to\infty} V(\Delta,X)/\log X\), then the better result \(\ell\geq\gamma_ 0/\pi+2\times 10^{-248}\) was obtained by the reviewer [Acta Arith. 59, 37-58 (1991; Zbl 0736.11048)] (without any hypothesis).
reviewed by: J. Kaczorowski
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 62237
zbMATH Keywords: distribution of primes
zbMATH Keywords: oscillations
zbMATH Keywords: number of sign-changes
zbMATH Keywords: remainder term
zbMATH Keywords: prime number theorem
MaRDI profile type: Publication
cites work: On sign-changes in the remainder-term of the prime-number formula, I
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cites work: On the number of sign changes in the remainder-term of the prime-ideal theorem
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cites work: Über Vorzeichenwechsel einiger arithmetischer Funktionen. II. (On sign- changes of certain arithmetical functions. II)
cites work: Q5803461
description: scientific article; zbMATH DE number 62237