On a number theoretic series
type: Item
instance of: scholarly article
title: On a number theoretic series
zbMATH Open document ID: 0659.10054
DOI: 10.1016/0022-314X(88)90011-X
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: The series \[ S(x)=1+\sum_{k\geq 1}\frac{1}{k!}\sum_{n_ 1n_ 2...n_ k\leq x;\quad n_ 1,n_ 2,...,n_ k>1}\frac{1}{\log n_ 1\cdot \log n_ 2\cdot \cdot \cdot \log n_ k} \] for \(x>1\), where \(n_ 1,n_ 2,...,n_ k\) are ordinary integers is estimated by means of Vinogradov's trigonometric sum method and contour integration. Improving the result of \textit{H. G. Diamond} [Pac. J. Math. 111, 283-285 (1984; Zbl 0541.10042)] \[ S(x)=c\cdot x+O(x \exp (-(\log x)^{1/2-\epsilon})) \] the author proves \[ S(x)=c\cdot x+O(x \exp (-c_ 1(\log x)^{3/5} (\log \log \log x)^{2/5})), \] where c and \(c_ 1\) are positive constants.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11M99
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 4077363
zbMATH Keywords: asymptotic results
zbMATH Keywords: number theoretic series
zbMATH Keywords: Vinogradov's trigonometric sum method
zbMATH Keywords: contour integration
author: Wenbin Zhang
MaRDI profile type: Publication
cites work: Asymptotic distribution of Beurling's generalized integers
cites work: A number theoretic series of I. Kasara
cites work: Q4071794
cites work: Q3328634
cites work: Q5803461
description: scientific article; zbMATH DE number 4077363