The main purpose of this paper is to use the mean value theorems of Dirichlet $L$-function to study the distribution properties of the generalized Dedekind sums, and gives two interesting asymptotic formulas.
References
T. M. Apostol, Theorems on generalized Dedekind sums, Pacific Journal of Mathematics, 2 (1952), 1-9. 0047.04502 10.2140/pjm.1952.2.1 T. M. Apostol, Theorems on generalized Dedekind sums, Pacific Journal of Mathematics, 2 (1952), 1-9. 0047.04502 10.2140/pjm.1952.2.1
L. Carlitz, Some theorems on generalized Dedekind sums, Pacific Journal of Mathematics, 3 (1953), 513-522. 0057.03701 10.2140/pjm.1953.3.513 L. Carlitz, Some theorems on generalized Dedekind sums, Pacific Journal of Mathematics, 3 (1953), 513-522. 0057.03701 10.2140/pjm.1953.3.513
L. J. Mordell, The reciprocity formula for Dedekind sums, Amer. J. Math., 73 (1951), 593-598. 0042.27401 10.2307/2372310L. J. Mordell, The reciprocity formula for Dedekind sums, Amer. J. Math., 73 (1951), 593-598. 0042.27401 10.2307/2372310
W. Zhang, On the mean values of Dedekind sums, Journal de Theorie des Nombres, 8 (1996), 429-442. 0871.11033 10.5802/jtnb.179 W. Zhang, On the mean values of Dedekind sums, Journal de Theorie des Nombres, 8 (1996), 429-442. 0871.11033 10.5802/jtnb.179
W. Zhang, A note on the mean square values of Dedekind sums, Acta Mathematica Hungarica, 86 (2000), 275-289. MR1756252 0963.11049 10.1023/A:1006724724840 W. Zhang, A note on the mean square values of Dedekind sums, Acta Mathematica Hungarica, 86 (2000), 275-289. MR1756252 0963.11049 10.1023/A:1006724724840
J. B. Conrey, E. Fransen, R. Klein and C. Scott, Mean values of Dedekind sums, J. Number Theory, 56 (1996), 214-226. 0851.11028 10.1006/jnth.1996.0014 J. B. Conrey, E. Fransen, R. Klein and C. Scott, Mean values of Dedekind sums, J. Number Theory, 56 (1996), 214-226. 0851.11028 10.1006/jnth.1996.0014
M. Xie and W. Zhang, On the Mean of the Square of a Generalized Dedekind Sum, The Ramnujan Journal, 5 (2001), 227-236 0997.11075 10.1023/A:1012954025283 M. Xie and W. Zhang, On the Mean of the Square of a Generalized Dedekind Sum, The Ramnujan Journal, 5 (2001), 227-236 0997.11075 10.1023/A:1012954025283