An \(\Omega\)-theorem for an error term related to the sum-of-divisors functions
type: Item
instance of: scholarly article
title: An \(\Omega\)-theorem for an error term related to the sum-of-divisors functions
zbMATH Open document ID: 0606.10035
DOI: 10.1007/BF01630684
published in: Monatshefte für Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178322
review text: Let \(\sigma\) denote the sum of divisors function, and set \(E_ 1(x):=\sum_{n\leq x}\sigma (n)-(\pi^ 2/12)x^ 2.\) Gronwall and Wigert proved (independently) in 1913/14 that \(E_ 1(x)=\Omega (x \log \log x).\) In this paper we obtain the more precise \(E_ 1(x)=\Omega_-(x \log \log x).\) The method consists in averaging \[ E_{- 1}(x):=\sum_{n\leq x}(\sigma (n)/n)-((\pi^ 2/6)x-(1/2) \log x) \] over suitable arithmetic progressions, and was suggested by the work of \textit{P. Erdős} and \textit{H. N. Shapiro} [Can. J. Math. 3, 375-385 (1951; Zbl 0044.039)] on the error term corresponding to Euler's function, \(\sum_{n\leq x}\phi (n)-(3/\pi^ 2)x^ 2.\)
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3979173
zbMATH Keywords: error term
zbMATH Keywords: Omega estimate
zbMATH Keywords: sum of divisors function
author: Y.-F. S. Pétermann
MaRDI profile type: Publication
cites work: On the Changes of Sign of a Certain Error Function
cites work: Changes of sign of error terms related to Euler's function and to divisor functions II
cites work: Über eine Vermutung von S. Chowla und H. Walum
description: scientific article; zbMATH DE number 3979173