Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2009; 46(1): 187-213

Printed January 1, 2009

Copyright © The Korean Mathematical Society.

On the mean values of Dedekind sums and Hardy sums

Huaning Liu

Northwest University

Abstract

For a positive integer $k$ and an arbitrary integer $h$, the classical Dedekind sums $s(h, k)$ is defined by $$ s(h,k)=\sum^k_{j=1}\left(\left(\frac{j}{k}\right)\right) \left(\left(\frac{hj}{k}\right)\right), $$ where $$ ((x))=\left\{\begin{array}{ll}\displaystyle x - [x] - \frac{1}{2}, & \hbox{if $x$ is not an integer;}\\ \displaystyle 0, & \hbox{if $x$ is an integer.} \end{array}\right. $$ J. B. Conrey et al proved that $$ \mathop{\sum_{h=1}^k}_{(h,k)=1} s^{2m}(h,k)=f_m(k)\left(\frac{k}{12}\right)^{2m}+O\left(\left(k^{\frac{9}{5}} +k^{2m-1+\frac{1}{m+1}}\right)\log^3k\right). $$ For $m\geq 2$, C. Jia reduced the error terms to $\displaystyle O\left(k^{2m-1}\right)$. While for $m=1$, W. Zhang showed
$$\begin{aligned}
\mathop{\sum_{h=1}^k}_{(h,k)=1}s^{2}(h,k)=&\ \frac{5}{144}k\phi(k) \prod_{p^{\alpha}\parallel k}\left[\frac{\left(1+\frac{1}{p}\right)^2-\frac{1}{p^{3\alpha+1}}} {1+\frac{1}{p}+\frac{1}{p^2}}\right]\\ &\ +O\left(k\exp\left(\frac{4\log k}{\log\log k}\right)\right). \end{aligned} $$ In this paper we give some formulae on the mean value of the Dedekind sums and and Hardy sums, and generalize the above results.

Keywords: Dedekind sums, Hardy sums, mean value, asymptotic formula

MSC numbers: 11F20