On certain Hardy sums and their \(2m\)-th power mean
type: Item
instance of: scholarly article
title: On certain Hardy sums and their \(2m\)-th power mean
zbMATH Open document ID: 1064.11035
author: Huaning Liu
author: Wenpeng Zhang
published in: Osaka Journal of Mathematics
publication date: 2005-03-18T00:00:00Z
review text: The sums in the title are defined by \[ S(d,c)= \sum^{c-1}_{j=1} (-1)^{j+1+[dj/c]}. \] For fixed \(m\geq 2\) and sufficiently large \(k\) it is shown that \[ \sum^k_{h=1} S^{2m}(h, k)= g_m(k) k^{2m}+ O(k^{2m-1}), \] where \(g_m(k)\) is a Dirichlet series that can be evaluated in closed form in terms of the Riemann zeta function. This improves a result of the second author in which \(k\) was restricted to be a prime.
reviewed by: Tom M. Apostol
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 2147082
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2147082