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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 $x \ne 0$ 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.

11F20Dedekind eta function, Dedekind sums
11M41Other Dirichlet series and zeta functions
11P55Applications of the Hardy-Littlewood method
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