On the mean values of the homogeneous Dedekind sums and Cochrane sums in short intervals

Abstract

In this paper, we study the mean values of the homogeneousDedekind sums and Cochrane sums in short intervals1;p3and1;p4,and give some asymptotic formulae by using the mean values of the Dirich-let L-functions.x 1. IntroductionFor integersa and q > 0, the classical Dedekind sum is dened byS(a;q) =qXr=1rqarq;where((x)) =x [x]12; ifx is not an integer;0; ifx is an integer:The sum S(a;q) plays an important role in the transformation theory of theDedekind function (see [7] and Chapter 3 of [1] for details).J. B. Conrey, E. Fransen, R. Klein and C. Scott [3] studied the mean valuesof Dedekind sums and proved the following:Proposition 1.1.Suppose thatm is a given positive integer andk is anysuciently large integer. ThenkX 0h=1S2m(h;k) = fm(k)k122m+ Ok9=5 + k2m 1+1 =(m+1)log3 k;Received November 4, 2005; Revised September 10, 2007.2000 Mathematics Subject Classication.11F20.Key words and phrases. Dedekind sums, Cochrane sums, homogeneous, mean values.Research partially supported by the National Natural Science Foundation of China un-der Grant No.60472068 and No.10671155; Natural Science Foundation of Shaanxi provinceof China under Grant No.2006A04; and the Natural Science Foundation of the Education