ON THE MEAN VALUES OF DEDEKIND SUMS AND HARDY SUMS

For a positive integer k and an arbitrary integer h, the classical Dedekind sums s(h, k) is defined by s(h,k) = (k)Sigma(j=1) ((j/k)) ((hj/k)), where ((x)) {x - [x] - 1/2, if x is not an integer; 0, if x is an integer. J. B. Conrey et al proved that (k)Sigma(h=1)s(2m)(h, k) = f(m)(k) (k/12)(2m) + O((k(9/5) + k(2m-1+1/m+1)) log(3) k). (h,w)=1 For m >= 2, C. Jia reduced the error terms to O (k(2m-1)). While for m = 1, W. Zhang showed (k)Sigma(h=1) s(2)(h,k) = 5/144k phi(k) Pi p alpha parallel to k [(1+1/p)(2) - 1/p(3 alpha+1)/1 + 1/p + 1/p(2)] (h,k)=1 + O (k exp (4 log k/log log k)) In this paper we give some formulae on the mean value of the Dedekind sums and and Hardy sums, and generalize the above results.