Mean values of Dedekind sums. (English) Zbl 0851.11028

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.


11F20 Dedekind eta function, Dedekind sums
11M41 Other Dirichlet series and zeta functions
11P55 Applications of the Hardy-Littlewood method
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