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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)= a=1 k a kah k

where ((x))=x-[x]-1/2 if x0 and 0 if x=0. The mean values of the 2mth 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

h=1 k-1 s(h,k) 2m =2·ζ(2m) 2 ζ(4m)k 12 2m +O(k 9/5 +k 2m-1+1/(m+1) ) log 3 k·

When k is a large integer, a similar estimate can also be obtained.


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