Journal of the Korean Mathematical Society (대한수학회지)

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ON THE MEAN VALUES OF DEDEKIND SUMS AND HARDY SUMS

  • Liu, Huaning (DEPARTMENT OF MATHEMATICS NORTHWEST UNIVERSITY)
  • Published : 2009.01.31
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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\limits_{j=1}^k\(\(\frac{j}{k}\)\)\(\(\frac{hj}{k}\)\),$$ where $$((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 $$\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\).$$ For $m\;{\geq}\;2$, C. Jia reduced the error terms to $O(k^{2m-1})$. While for m = 1, W. Zhang showed $$\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.

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References

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