On the mean value of Dedekind sums. (English) Zbl 0976.11044

For integer \(m\geq 1\) define \(f_m(k)\) be the Dirichlet series \[ \sum^\infty_{k=1} {f_m(k)\over k^s}=2 {\zeta^2(2m) \over\zeta (4m)}{\zeta (s+4m-1)\over \zeta^2(s+2m)} \zeta(s), \] and let \(s(h,k)\) denote the classical Dedekind sum. If \(\sum'\) indicates that \((h,k)=1\), the author derives the asymptotic formula \[ {\sum'}^k_{h=1} s^{2m}(h,k)=f_m(k) \left({k \over 12}\right)^{2m} +O(k^{2m-1}), \] for \(m\geq 2\) and sufficiently large \(k\). This improves a result by J. B. Conrey, E. Fransen, R. Klein, and C. Scott [J. Number Theory 56, 214-226 (1996; Zbl 0851.11028)] who had a much larger error term. For the case \(m=1\) there is a corresponding improvement with error term \(O(k^{3/2}\log^2k)\) in place of \(O(k^{9/5}\log^3k)\).


11N37 Asymptotic results on arithmetic functions
11F20 Dedekind eta function, Dedekind sums


Zbl 0851.11028
Full Text: DOI


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