Research On The Properties Of M-th Dedekind Sums And Quasi Dedekind Sums

Let h,q be integers with q>0 and(h,q)=1.This paper mainly generalizes the classical Dedekind sums.We define the m-th Dedekind sums S(h,m,q)which is a generalization of the classical Dedekind sums s(h,q),study square mean value of m-th Dedekind sums.And We define quasi Dedekind sums Cm(h,q)and Sm(h,q),study upper bound estimate of Cm(h,q)and square mean value of Sm(h,q).Then,The Rademacher's question on the Hardy sums is considered and we give some arithmetics1.Let p be an odd prime.This paper gives an asymptotic formula for mean square value of S(h,m,q)by using the relationship between m-th Dedekind sums and Dirichlet L function and a sharp asymptotic formula for a new mean value of the Dirichlet L-functions.For fixed positive odd integer m and p be a prime large enough,we have(?)2.By using estimates of exponential sums and some properties of character sums,we get the upper bound estimate of generalized Dedekind sums:(?) And using the relationship between Cochrane sums and Dirichlet L function,we give a sharp asymptotic formula for square mean value of Sm(h,q)3.Let a,b,x,y be positive integers with ay-bx=±1 and k?N.Some arithmetics on the Hardy sums Si(x+ak,y+bk)(i=1,2,3,4)are obtained This paper also considers zero points of Hardy sums.The Rademacher's question on the Hardy sums is answered.