On a sum involving the divisor function
type: Item
instance of: scholarly article
title: On a sum involving the divisor function
zbMATH Open document ID: 1499.11293
author: Jing Ma
author: Huayan Sun
published in: Periodica Mathematica Hungarica
publication date: 2021-11-27T00:00:00Z
review text: The authors use some newer techniques to show that the following estimate is valid for the divisor function \(d\): \[\sum_{n \leq x} d\left(\left[\frac{x}{n}\right]\right)=x \sum_{m \geq 1} \frac{d(m)}{m(m+1)}+O_{\varepsilon}\left(x^{11 / 23+\varepsilon}\right)\quad (x \rightarrow \infty).\] A part of the results of the paper that is needed in the course of the proof and is interesting in itself, gives estimates for \[\mathfrak{S}_{\delta}(x, D):=\sum_{D<k \leq 2 D} d(k) \psi\left(\frac{x}{k+\delta}\right)\quad(x \geq 2,\, 1 \leq D \leq x),\] where \(\psi(t):=t-[t]-\frac{1}{2}\), and \(\delta \in\{0,1\}\).
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 7434714
zbMATH Keywords: divisor function
zbMATH Keywords: asymptotic formula
zbMATH Keywords: exponent pair
reviewed by: István Mező
MaRDI profile type: Publication
full work available at URL: s10998-020-00378-3
OpenAlex ID: W3131735788
cites work: On a sum involving the Euler function
cites work: Q4692762
cites work: On a sum involving the Mangoldt function
cites work: On a sum involving the Euler totient function
cites work: Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski
DOI: 10.1007/S10998-020-00378-3
description: scientific article; zbMATH DE number 7434714