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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
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/ 1972 / The Mathematical Association of America
/ 1976 / Classical theorems on quadratic residues : 3 ~ 4
/ 1976 / Modular functions and Dirichlet series in number theory GraduateTexts in Mathematics
/ 1982 / An exact formula for an average of L-series illi (is j)
/ 1996 / Mean values of Dedekind sums 56 (2) : 214 ~ 226
/ 1976 / Classical theorems on quadratic residues : 3 ~ 4
/ 1976 / Modular functions and Dirichlet series in number theory GraduateTexts in Mathematics
/ 1982 / An exact formula for an average of L-series illi (is j)
/ 1996 / Mean values of Dedekind sums 56 (2) : 214 ~ 226
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