ON THE MEAN VALUES OF DEDEKIND SUMS AND HARDY SUMS

Title & Authors
ON THE MEAN VALUES OF DEDEKIND SUMS AND HARDY SUMS
Liu, Huaning;

Abstract
For a positive integer k and an arbitrary integer h, the classical Dedekind sums s(h,k) is defined by $\small{S(h,\;k)=\sum\limits_{j=1}^k$$\(\frac{j}{k}$$\)$$\(\frac{hj}{k}$$\),}$ where $\small{((x))=\{{x-[x]-\frac{1}{2},\;if\;x\;is\;not\;an\;integer; \atop \;0,\;\;\;\;\;\;\;\;\;\;if\;x\;is\;an\;integer.}\}$ J. B. Conrey et al proved that $\small{\sum\limits_{{h=1}\atop {(h,k)=1}}^k\;s^{2m}(h,\;k)=fm(k)\;$$\frac{k}{12}$$^{2m}+O$$\(k^{\frac{9}{5}}+k^{{2m-1}+\frac{1}{m+1}}$$\;\log^3k\).}$ For $\small{m\;{\geq}\;2}$, C. Jia reduced the error terms to $\small{O(k^{2m-1})}$. While for m = 1, W. Zhang showed $\small{\sum\limits_{{h=1}\atop {(h,k)=1}}^k\;s^2(h,\;k)=\frac{5}{144}k{\phi}(k)\prod_{p^{\alpha}{\parallel}k}$\frac{$$1+\frac{1}{p}$$^2-\frac{1}{p^{3\alpha+1}}}{1+\frac{1}{p}+\frac{1}{p^2}}$\;+\;O$$k\;{\exp}\;\(\frac{4{\log}k}{\log\log{k}}$$\).}$. 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;
Language
English
Cited by
1.
A new generalization of Hardy–Berndt sums, Proceedings - Mathematical Sciences, 2013, 123, 2, 177
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