Explicit bound of the divisor function
type: Item
instance of: scholarly article
title: Explicit bound of the divisor function
zbMATH Open document ID: 1484.11191
author: Thomas Dubbe
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2021-05-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1588903263
review text: The author proves that the sum of values of Piltz divisor function up to \(x\) satisfies the upper bound \((x(\log x + C_k)^{k-1})/(k-1)!\) for \(1\le x<3\) and \(k\ge 3\) or \(x\ge 3\) and \(k\ge 2\) , where \(C_k\) is approximately \(0.807k -1\). This improves the famous old upper bound by \textit{K. Mardzhanishvili} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 22, 387--389 (1939; Zbl 0021.20802)], where in place of \(C_k\) appears \(k-1\). The interesting elementary method is based on mathematical induction. As application, one can obtain improved explicit upper bounds for the class number of number fields.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 7344691
zbMATH Keywords: divisor function
zbMATH Keywords: explicit bound
zbMATH Keywords: Piltz divisor function
reviewed by: József Sándor
MaRDI profile type: Publication
OpenAlex ID: W3021255305
cites work: Q5433733
cites work: Algorithms in Algebraic Number Theory
cites work: Q5774060
cites work: Upper bounds for sums of powers of divisor functions
cites work: A Remark on Stirling's Formula
cites work: 75.9 Euler’s Constant
DOI: 10.7169/FACM/1854
description: scientific article; zbMATH DE number 7344691