# zbMATH — the first resource for mathematics

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.

##### 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
Full Text: