Conrey, J. B.; Fransen, Eric; Klein, Robert; Scott, Clayton Mean values of Dedekind sums. (English) Zbl 0851.11028 J. Number Theory 56, No. 2, 214-226 (1996). For a positive integer \(k\) and an arbitrary integer \(h\), the Dedekind sum \(s(h,k)\) is defined by \[ s(h,k) = \sum^k_{a = 1} \left( \left( {a \over k} \right) \right) \left( \left( {ah \over k} \right) \right) \] where \(((x)) = x - [x] - 1/2\) if \(x \neq 0\) and 0 if \(x = 0\). The mean values of the \(2m\)th moment of \(s(h,k)\) is investigated via the circle method of Hardy and Littlewood in this paper. Suppose that \(k\) is a large prime number; then it is proved that \[ \sum^{k - 1}_{h = 1} s(h,k)^{2m} = 2 \cdot {\zeta (2m)^2 \over \zeta (4m)} \left( {k \over 12} \right)^{2m} + O \bigl( (k^{9/5} + k^{2m - 1 + 1/(m + 1)}) \log^3k \bigr). \] When \(k\) is a large integer, a similar estimate can also be obtained. Reviewer: Pei Dingyi (Beijing) Cited in 10 ReviewsCited in 38 Documents MSC: 11F20 Dedekind eta function, Dedekind sums 11M41 Other Dirichlet series and zeta functions 11P55 Applications of the Hardy-Littlewood method Keywords:Dedekind sum; mean values; circle method PDF BibTeX XML Cite \textit{J. B. Conrey} et al., J. Number Theory 56, No. 2, 214--226 (1996; Zbl 0851.11028) Full Text: DOI arXiv