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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 \(O(k^{9/5}\log^3k)\).

MSC:

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

Citations:

Zbl 0851.11028
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References:

[1] Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0332.10017
[2] Conrey, J. B.; Fransen, E.; Klein, R.; Scott, C., Mean values of Dedekind sums, J. Number Theory, 56, 214-226 (1996) · Zbl 0851.11028
[3] Hall, R. R.; Huxley, M. N., Dedekind sums and continued fractions, Acta Arith., 63, 79-90 (1993) · Zbl 0785.11027
[4] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1979), Oxford University Press: Oxford University Press Oxford · Zbl 0423.10001
[5] Walum, H., An exact formula for an average of \(L\)-series, Illinois J. Math., 26, 1-3 (1982) · Zbl 0464.10030
[6] Zhang, W., On the mean values of Dedekind sums, J. Théor. Nombres Bordeaux, 8, 429-442 (1996) · Zbl 0871.11033
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