A Study On The Mean Of New Dedekind Sums

Let f,k be positive integers,h be arbitrary integer,satisfying(fk,h)=1 and(f,k)=1.The Bernoulli numbers Bn and Bernoulli polynomials Bn(x)are defined as followsLet x be a primitive character of modulus k.The generalized Bernoulli polyno-mials Bn,x(x)are defined by means of the generating functionLet (?)n(x)be the n-th Bernoulli character function such that{Bn(x)={Bn(x-[x],if x is not an integer,0,if x is an integer.Let (?)n,x(x)denote the n-th generalized Bernoulli character function satisfying{Bn,x(x-[x],if x is not an integer,0,if x is an integer.Define new Dedekind sums as following:S(h,k;x_=fk-1?a=B1,x(ha/k)B1(a/fk).First,this paper established some connection between S(h,k;x)and Dirich-let L-function.Then,we gave asymptotic formula for the second power mean of S(h,k;x)and the hybrid mean of S(h,k;x).