Changes of sign of error terms related to Euler's function and to divisor functions
type: Item
instance of: scholarly article
title: Changes of sign of error terms related to Euler's function and to divisor functions
zbMATH Open document ID: 0613.10037
DOI: 10.1007/BF02621904
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140042
review text: The author considers the sign change problem of the error terms R(x) (resp. H(x), \(F_ k(x)\), \(P_ k(x))\) associated with the Euler function \(\phi\) (n) (resp. with \(\phi\) (n)/n, the sum-of-divisors function \(\sigma_ k(n)\), the lattice point counting function in certain 4- dimensional ellipsoids). He proves two main theorems which give lower bounds for the number \(X_ f(x)\) of changes of sign of the real function f (taken to be one of the above error functions). Theorem 1 gives the estimate \[ X_ f(x)\geq const\cdot x+o(x) \] under the condition that \(f(x)=f(n)-C\{x\}+o(1)\) between integers n and \(n+1\) and the mean square integral of f is bounded above by \(Kx+o(x)\) with constants \(C>0\) and \(K\geq 0\). This is applied to \(f=H\), \(F_ k\) \((k<0)\) on the basis of the mean value theorems of Chowla and Walfisz dating back to 1930's. To cover the cases \(f=R\), \(F_ k\) \((k>0)\), \(P_ k\) the author proves Theorem 2: If f is as in Theorem 1 and real functions g,h \((h>0)\) satisfy \[ g(x)=h(x)(f(x)+o(1)), \] then one has the same lower bound for \(X_ g(x)\). If, in addition, the distribution function \(D_ f\) for f exists and is continuous, then one gets the asymptotic identities in the conclusions of Theorems 1 and 2. Using the result of \textit{P. Erdős} and \textit{H. N. Shapiro} [Can. J. Math. 7, 63-75 (1955; Zbl 0067.276)] on the existence and continuity of \(D_ H\), the author gets asymptotic identities for \(X_ H\) and \(X_ R.\) The paper is well written and pleasant to read, with carefully made (historical) remarks. In particular, the author is the first to notice that Sylvester has unconsciously disproved his own conjecture on R(n).
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 3991611
zbMATH Keywords: number of lattice points
zbMATH Keywords: sign change
zbMATH Keywords: error terms
zbMATH Keywords: Euler function
zbMATH Keywords: sum- of-divisors function
zbMATH Keywords: 4-dimensional ellipsoids
author: Y.-F. S. Pétermann
reviewed by: Shigeru Kanemitsu
MaRDI profile type: Publication
OpenAlex ID: W3177314736
description: scientific article; zbMATH DE number 3991611