The mean value theorem of the divisor problem for short intervals
type: Item
instance of: scholarly article
title: The mean value theorem of the divisor problem for short intervals
zbMATH Open document ID: 0926.11072
author: Isao Kiuchi
author: Yoshio Tanigawa
published in: Archiv der Mathematik
publication date: 1999-12-02T00:00:00Z
review text: Let \(-1< a\leq 0\) and let \(\sigma_a(n)\) denote the sum of the \(a\)th powers of positive divisors of \(n\). It is considered the error term \(\Delta_a(x;r)\) in the asymptotic representation of \(\sum_{n\leq x}\sigma_a(n) e^{2\pi irn}\), where \(r= h/k\) with coprime \(h,k\). It is proved a complicated asymptotic expansion of the mean value \[ \int_X^{X+H}| \Delta_a(x+U;r)- \Delta_a(x;r)|^2 dx \] for \(X\geq 2\), \(1\leq U\ll x^{1/2}\ll H\leq X\), \(4kU\leq X\).
reviewed by: Ekkehard Krätzel
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1292435
zbMATH Keywords: mean value theorem
zbMATH Keywords: short intervals
zbMATH Keywords: asymptotic results
zbMATH Keywords: divisor functions
zbMATH Keywords: error term
zbMATH Keywords: asymptotic representation
MaRDI profile type: Publication
full work available at URL: s000130050289
OpenAlex ID: W2047241980
DOI: 10.1007/S000130050289
description: scientific article; zbMATH DE number 1292435