A sum analogous to the Dedekind sum and its mean value formula
type: Item
instance of: scholarly article
title: A sum analogous to the Dedekind sum and its mean value formula
zbMATH Open document ID: 0997.11076
published in: Journal of Number Theory
publication date: 2002-06-04T00:00:00Z
review text: Let \(((x))= x-[x]- \frac{1}{2}\) if \(x\) is not an integer, and let \(((x))=0\) otherwise. The sum in the title, originally introduced by Hardy, is defined by \[ S_2(h,k)= \sum_{a=1}^k (-1)^a \biggl(\biggl( \frac{a}{k} \biggr)\biggr) \biggl(\biggl( \frac{ah}{k} \biggr)\biggr). \] For even \(k\) it can be expressed in terms of the classical Dedekind sum \(S(h,k)\) by the relation \[ S_2(h,k)= 2S(h,k/2)- S(h,k). \] The author uses some of his earlier work on \(S(h,k)\) to deduce corresponding results for \(S_2(h,k)\). A corollary of the main theorem gives, for an odd prime \(p\), \[ \mathop{{\sum}'}_{h=1}^{2p} |S_2(h,2p)|^2= \frac{1}{32} p^2+ O\Biggl(p\exp \biggl( \frac{4\log p}{\log\log p} \biggr)\Biggr). \]
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11F20
zbMATH DE Number: 1749020
zbMATH Keywords: mean value formula
zbMATH Keywords: asymptotic result
zbMATH Keywords: Hardy sum
zbMATH Keywords: Dedekind sum
author: Wenpeng Zhang
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
full work available at URL: jnth.2000.2624
OpenAlex ID: W2029885858
cites work: Q4101884
cites work: Q4097001
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DOI: 10.1006/JNTH.2000.2624
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description: scientific article; zbMATH DE number 1749020